If you do the bleeding on your own, and you get burned or hurt somehow, what then? Are you going to sue the landlord? Can you afford the lawyers? Do you HAVE a case? (I mention this because, well obviously, if you knew how to bleed radiators, you wouldn't ask how it's done.)
What I suggest you do now, is work your way 'backwards' and document your converstaions (including conversations you had in person AND on the phone), with dates, AND times, AND what you AND he said to each other. The next thing it to send him a certified letter, w/a RETURN RECEIPT that reiterates what the BOTH of you said to each other in the past, with dates, and by what means, and tell him Winter is coming. And state which radiators are working and not working, still. And acknowledge what he's already done and what he said he'd do. THEN, tell him he's already 'promised' to fix the problem a couple of times (or is it once), and if it doesn't get done soon, you'll have it done by a PROFESSIONAL and will deduct the expense from your rent. Give him a few reasons why it's a 'desperate situation'. (You have small children ..., or a short-haired dog ..., or you live with your 80 year old mother who has poor blood circulation ..., and/or so on ...)
The way it works is, IF you ever end up in front of a judge over all this, you will then have the NECESSARY documentation to back it up, i.e., your Log Book. All the judge is looking for is that you gave the Landlord REASONABLE time to fix the problem. By doing it yourself, outright, or having a pro do it outright, the Landlord can argue that you didn't give him enough opportunity to fix the radiators. (He already made one attempt and said he'd do it again.) You see, he's already done it once and it's only fair to have him try again. BUT, you must give him opportunity. Just document all conversations/meetings/phone calls/'parking lot encounters'/maintenance-man-encounters/and-so-on. ONCE you've exhausted all REASONABLE means for him to fix the problem, then when the apartment's temperature reaches 55 degrees, you call him to tell him he has 24 hours to fix the heat (and you document this). Then, if he doesn't say when it's going to be done, you could reasonably FOLLOW-THROUGH on your 'alternative' (to have a pro do it, and deduct the expense from the rent.)
So again, in the meantime, document the past, bug him again today (and document it), and bug him as often as reasonably possible (and document these too.) Be SURE to ALWAYS ask him when he's gonna do it so you can be there. (Otherwise, he can say he did it and it never got done.) AND you can prove he said he'd fix it on XYZ date and didn't. At some point, if he continually fails to make the fix, you need a pro. Just DON'T do this yourself. You need your Landlord's permission and he will want to absolve himself of all liability. I can bet he simply wants to do it himself ..
Oh, and one last thing ... Remember, if you end up having a pro do the work and end up deducting the bill from your rent, then don't be surprised the next time your lease comes up that he 'gets even' with you, or worst, doesn't renew. That's my advise. My best to ya and hope this helps.