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I can't recall any other episode which specifically deals with mental health.

Star Trek is full of plots that deal with mental health. I think just about every episode contains some discussion around mental health in one form or another. Here are a few examples of plots that span multiple episodes or movies:

• 
  Deanna Troi serves as ship's counselor, and we see several crew members attending therapy.


• The crew of Voyager deals with being so far away from home.


• 
  Benjamin Sisko copes with the loss of his wife.


• 
  Julian Bashir comes to grips with being outed as a genetically modified person.


• 
  Worf struggles with being the first Klingon in Starfleet.


• 
  Jean-Luc Picard deals with his assimilation (it's a major plot point in First Contact) and the death of his brother and nephew.


• 
  Data strives to feel emotions, and to understand what it's like to be human.

Is there any reference to a character struggling with a long-term mental illness that is openly discussed on the show?

The first character that comes to mind is Reginald Barclay.

From Wikipedia, emphasis mine:

While possessing great technical skill and sincere enthusiasm, Barclay seems anxiety-ridden, socially awkward, and self-conscious. He also displays stuttering and dysfluent speech behaviors, especially when nervous or anxious, along with some secondary gestures, such as facial grimaces and small head tics, however on the holdeck many of these issues vanish and he acts confident. He has an obsessive interest in fantasy, which seems to serve as an escape from personal interactions, especially those where he is being mistreated by others. Barclay's anxieties extend to idiosyncratic fears (such as fear of being transported) and hypochondriasis. The overcoming of his fears and social anxieties became a running plot point across many seasons in multiple Star Trek series.

Miles O'Brien also has a few story lines related to his mental health.

Again, from Wikipedia:

The TNG episode "The Wounded" establishes that O'Brien served as tactical officer aboard the USS Rutledge during the Cardassian War and that he was emotionally scarred by the Cardassians' massacre of hundreds of civilians on Setlik III.

...

The producers would routinely put O'Brien under intense psychological pressure in episodes jokingly dubbed 'O'Brien must suffer'.

...

Miles was temporarily relieved of duty after a visit to the planet Argratha. O'Brien had been falsely accused of espionage and was given the simulated memory of a 20-year prison sentence. O'Brien became paranoid and emotionally distressed upon his return to the station and attempted suicide (Episode: "Hard Time").

TOS had one. "The Doomsday Machine" had Commodore Matt Decker lose his crew despite taking the action he believed would save them. He comes away with post traumatic stress (before it even had a name) and guilt as both survivor and inadvertent agent of the death of his entire crew. It all manifests as he bullies his way into temporary command of the Enterprise to attempt a reckless and futile attack on the planet eater. When Spock relieves him of command, he steals a shuttlecraft in which he gives up his own life.

DS9 attempts to tackle PTSD in the episode "It's Only A Paper Moon" (season 7, episode 10).

The plot revolves around Nog dealing with the fallout from losing his leg in a previous episode. I'm not at all qualified to comment on how well the episode does at handling PTSD, but, being an episodic show, it's still mostly resolved by the end of the episode.

Edit to add that you could make an argument that in the DS9 episode The Visitor (season 4, episode 3), it shows Jake dealing with the long term effects of grief, though they don't really address it head on.

The same can be said of Janeway in the season 7, episode of Voyager "The Void." That episode shows her coming to grips with her guilt over stranding Voyager in the Delta Quadrant.

Several TNG episodes were centered around various mental health problems, mostly (though not always) of minor characters.

Perhaps most prominently, 4x02 "Family" deals with Captain Picard's lingering trauma from his capture by the Borg over the course of "The Best of Both Worlds". This event and its long-term impact will continue to be referenced throughout the series and into the movie First Contact.

4x12 "The Wounded" centers around Captain Maxwell, a veteran of the war with the Cardassians, who suffers some flavor of PTSD and is convinced as a result that the Cardassians need to be destroyed. (I'd call this paranoia, except that they were indeed up to no good.)

Grief and coping were the subject of several episodes, including 5x11 "Hero Worship", where a child survivor of a wrecked ship tries to emulate Data in order to bury his grief, and 7x07 "Dark Page" that details Lwaxana Trio's (characteristically overwrought) reaction to the long-ago death of her first daughter.

Suicide was the topic of 7x18 "Eye of the Beholder", with the unexpected and initially unexplained death of a junior Enterprise officer. Bonus Lwaxana points: she also had a brush with suicide in 4x22 "Half a Life", although that was socially-expected euthanasia rather than a mental issue per se.

I can recall two episodes about psychotic disorders that try to look deeper into uncontrollable urge for violence.

One is "Meld" from "Voyager" - about a member crew Suder who killed people "for no reason" and Tuvok, being unable to understand this illogical action tries to realize what such psychotic perception really is.

In "Repentance" ("Voyager" again) they transport a convicted murderer who is explained to be a typical psychopath, feeling joy harming others. He is a violent man who did awful things. The Doctor examines him and concludes that he got a neuro-physiological explanation to his mental state. The Doctor then is able to fix his brain physiology to make him able to feel regret and empathy, which he was never able to feel before. After that they face a moral problem - can the "new person" who was cured of his mental illness be judged and punished for what he did previously.

In some ways, the TOS episode Dagger of the Mind dealt with this. Dr. Tristan Adams invents a neural neutralizer that can render a subject susceptible to suggestions. In his experiments to best use the device, he drives one of his own staff mad. He also produces people devoid of any emotions. The device ultimately kills him when it reactivates unexpectedly and empties Adams' mind of everything, since nobody was there to replace what the device was taking away.

The (perhaps shortsightedly written) TOS episode Whom Gods Destroy involves an asylum for the criminally insane, multiple criminally insane characters including Garth of Izar, as well as a revolutionary medicine which was believed to have the ability to wipe out any and all remaining forms of mental illness.

"The Outcast" in Star Trek:TNG has Ryker become romantically involved with Soren, a J'naii. The androgynous J'naii see all expressions of sexuality as mental illness. Soren is discovered and successfully (by J'naii standards) treated.

I'll throw in the obvious TOS Amok Time. Spock goes off the rails and the title of the episode "Amok" is a Malay word for going nuts and berzerkoid.

In the TOS episode The Ultimate Computer the famous scientist Dr. Richard Daystrom (namesake of the Daystron Institude) has a mental breakdown after the M5 incident, and needed to be taken to "a total rehabilitation facility". The condition does not appear to be purely physical, thus the need for total rehabilitation.

Act Three

[...]In the briefing room, the senior staff collaborates on a plan to gain back control by focusing on a certain relay unit between the M-5 and the bridge. McCoy goes to Daystrom to convince him to shut off the M-5. Daystrom, on the other hand, defends the M-5, saying it's learning, and further, that the advance the M-5 represents would liberate man from hazardous duties, saving life. McCoy notes later to Kirk that Daystrom reacts toward the computer as a father would to his child. Even if the child went anti-social and killed a person, a father would protect the child.

Act Four

[...]Now that the M-5 has committed murder, Kirk confronts Daystrom, convincing him that the M-5 is doing more than originally designed. He demands that Daystrom attempt to reason with M-5, as Daystrom admits it is his engrams that he imprinted on the machine. However, he goes mad in the effort, realizing his reputation is at stake. In his delirium, he lashes out at Kirk, but is subdued by a Vulcan nerve pinch from Spock.

McCoy hauls him off to sickbay, and Spock notes the self-preservation that the M-5 is displaying is probably a consequence of Daystrom's engram imprinting.

[...]Dr. Daystrom, meanwhile, is cared for in sickbay under sedation and heavy restraint to await transfer to a total rehabilitation facility, under McCoy's recommendation. Kirk orders that Sulu plot a return course to Starbase 6.

Source: http://memory-alpha.wikia.com/wiki/The_Ultimate_Computer_(episode)

Doctor Richard Daystrom was one of the most influential Human scientists of the 23rd century. Born in 2219, Daystrom was considered a genius in his day, and was compared to Albert Einstein, Kazanga, and Sitar of Vulcan. He was the inventor of the comptronic and duotronic computer systems.

Source: http://memory-alpha.wikia.com/wiki/Richard_Daystrom