I don't know the nature of how he became your "mentor", but a mentor does not have to be a standardized thing with formal schedules, expectations of LOR, etc. The poster I quoted below captures a lot of what is great about a mentor.
To me, it would seem odd to even ask for a LOR from such a mentor if you haven't had them for a class since your relationship will essentially center around conversations and email correspondence. The best value from a mentor won't be a LOR simply in learning the way the world works (from his perspective) and getting advice from someone that's seen it and done it.
I also feel compelled to point out that just because someone wants to be your mentor doesn't mean they'll be a good one. A lot of what makes a good mentor will depend on how good you two's chemistry is, how open and frank he is with you, how broad his experiences and knowledge is such that he can advice you on a wide range of topics, how much time he actually has to devote to you, and how willing he is to network for you, make calls, send letters of introduction, etc.
Think of it like college advisors. You can see them any time, or perhaps you only see them on scheduled appointments. Perhaps they know a lot about a lot of different majors, departments, professors, and classes--while they may only know a lot about their own department. Just because someone agrees to be an advisor (like a premed advisor for example) doesn't mean they'll be good at it or a good fit for you.
Mentoring is awesome though. I'm still looking for that great mentor. I've had a lot of really good ones that have addressed specific aspects of my life and future career goals but not one that really brings it all together and has been there and seen what I want to do. Mentoring can be awesome stuff when you make it work.