Yep, Mikado's been working there for... about half a year now? He works in the electronic library, but he also does sometime go to the gym, and would be around public spaces like the cafeteria. If Mikoto encountered him at the gym being ridiculous about exercise - he tends to try to overdo it for his fitness level - that might be a good "in" to Mikado actually opening up about his problems?

It's a fun series! I actually play from pretty far into the light novel series, so it's a little different from the anime, but the anime is tons of fun and I would definitely recommend it to anyone!