I know a few dramas more or less like that!
Gomen ne, seishun
http://mydramalist.com/10860-gomenne-seishun male teacher!
Hi wa mata noboru
http://mydramalist.com/2757-hi-wa-mata-noboru I liked the mentor/student thing in this one. (I'm pretty sure I wrote a review once but it's gone from the page!)
Koukou kyoushi 2003
http://mydramalist.com/790-kou-kou-kyoushi-2003 Male teacher/student - intense and kind of twisted story but sooo good. You don't have to like what the guy is doing to feel for him!
My rainy days
http://mydramalist.com/764-my-rainy-days Taisetsu na koto wa subete kimi ga oshiete kureta
http://mydramalist.com/10-taisetsu-na-koto-wa-sube...Beethoven Virus
http://mydramalist.com/135-beethoven-virus The male lead looks like a complete jerk but you'll soon see that there's more beneath the surface; there's a mentor/student romance going on as well as a mentor/student love/hate bromance! (Don't be bothered by the rating, trust me, it's way better than that! I suspect it's under rated because some fangirls like the younger guy better.)
Koi
http://mydramalist.com/8320-koi it's more about the romance (love triangle but it's a GOOD one!) than the mentor/student thing though, even though that's how it starts.
Valid Love
http://mydramalist.com/10874-valid-love Romance, love triangle, the male lead is a teacher but most of the drama after the first two episodes is about their life years later. (So, maybe the weakest recommendation on this list)
Dinner
http://mydramalist.com/5998-dinner The male lead is a chef who comes to help a restaurant in trouble, he's like a grumpy knight in shining armour who saves the sweet princess (the romance is very subtle and between the lines though or perhaps only in my head...)
Galcir
http://mydramalist.com/557-galcir kind of weird comedy, the lead acts sort of like an alien at times but he's not... but he's acting like a kind of mentor-figure to the silly girls. Maybe you won't like it, it's very much "only j-drama can be this silly!!!"