There are several reasons why you should not contact your manager:
You're off sick. I'm going to assume some sort of insurance is involved here. Sometimes the "time off" is guaranteed only because you are unable to perform your duties. An insurance company might claim that because you are able to work, they shouldn't have to support some of your expenses, etc. Analyze your situation from this point of view.
You've taken a not insignificant time off of work. I'm sure that it's impacted your office quite heavily. The only reason so much time off would be approved is for a very serious medical reason, with the understanding that you simply cannot fulfill your duties. And then your manager gets an email saying:
"Hi, I could do some work. But not all of it. And not full work-days, or maybe not even during business hours. In fact, give me something to do and I'll chip away at it. I'll finish it soon. Maybe. If I feel well. If."
Your manager/team/company may think you are not as sick as you claimed you were. This could be ruinous to your future in that company.
Consider the above text. Your manager might be in a bind with you gone, but what he needs most is not just that the job get done, but that it gets done reliably. You will, of course, express yourself far more professionally than I did up there, but what it comes down to is that you can't offer stability or reliability. You may feel well, or you may not. You may put 5 hours into a task today, or 2, because you're sick, or need to go to the doctor. You're not in a position to guarantee productivity, work hours, etc. Accepting your help would just throw an extra wrench in the manager's daily routine, as he wouldn't know when to expect your work to be done, or how to tell you to hurry up, since you're sick and he can't really ask anything of you.
4. Give them a finger ...
Your manager might be truly swamped due to your absence. They can't wait for you to get back. Then you volunteer your help. They thank you profusely, and assign you a minor task to get done. You feel useful, and chip away at it when you can. But then your other co-worker calls in sick.
They had a big deadline coming up, and them not finishing their task will set the whole project back. Well, bnjmn knows how to do that stuff, right? And he volunteered to help, it's not like we're in the wrong to ask. So bnjmn, there's this task .. not much needs to be done .. just fill in a few details .. can't be more than a couple of hours worth of work, and it's really urgent! Do you think you could help us out? Yes? Fabulous! Oh, can I get the revised version before the end of the day? OK, great, thanks.
Do you want to end up in that situation? Because you very well may.
5. Your health
You're off because you're recovering from a medical condition. Your "job" during this time is to get better. Relax! Read some books, learn something new, get involved with a hobby - within your current capabilities, of course. You'll be working your whole life!
Take the chance to rest and recuperate when it is offered to you - it's not like you don't have a good reason for it! Forget the office, forget work. Just focus on getting better.
That's my (long winded) 2 cents. Hope it helps, and get better!