I'm coming up with a monk subclass that wears Medium armor while still gaining the effect of Unarmored Defense, but I need to balance it correctly. Any suggestions?

For context, this subclass is supposed to fulfil the fantasy of "a samurai with fisticuffs" as the player asking for this put it. The subclass revolves around both tanking damage as well as dishing out heavy damage and additional effects on unarmed strikes. Why they didn't want to play a samurai fighter, i'll never know.

That's a pretty large AC bonus in practice. Even if it's Armor AC +dex (max 2) +wis, it's +2 AC over baseline monk, even before you take into account magic armor. If you allow the full dex bonus in medium armor, that's up to +5 before magic over baseline.

The most obvious balancing factor to try is if it doesn't make all the monk abilities armor-compatible, but then you're rapidly turning the monk into "punchy fighter".

Similarly, what's the subclass's ki economy supposed to be? What's competing with or enhancing their basic ki abilities?

The subclass revolves around both tanking damage as well as dishing out heavy damage and additional effects on unarmed strikes. Why they didn't want to play a samurai fighter, i'll never know.

This really feels like it's trying to push the monk into the fighter, paladin, or barbarian's role, and that's going to be difficult to do in a balanced way. The monk isn't tanky, and, while they do decent damage, they don't do heavy damage. They're a mobile striker, and the features that make them that are part of the main class.

Since this seems to be at a player request, you're allowed to go back to them and say "you ask too much". Whatever concept they're trying to realize, it may be better served by a fighter subclass.

If you want to allow armor on the monk I would say don’t allow unarmored defense at the same time. It’s one or the other, like Barbarian. But I might remove the “no armor or shield” restriction on Martial arts. 

edit: I wouldn’t allow shields though

Thanks for the input! I think that instead of allowing unarmored defense, I might just have it so the subclass gains alternate benefits when wearing armor, which could also open up heavy armor as an option. I don't think I'll allow shields though. Most monks wouldn't want a hand taken up anyways.