Obviously I don't know you or your friend but there are some things that strike me about this. I may be projecting my own situation onto it though! So take the following with a pinch of salt. :)
She has tried therapy and takes antidepressants but they don't seem to help.
I'm wondering what kind of therapy she had. Quite likely a short-term behavioural intervention rather than a longer term, get to the root of the problem type thing. Perhaps more therapy (probably private) might help. Antidepressants don't work for everyone, or don't necessarily work a huge amount when they do work. They are certainly not the magic pill some people think. (Not you Intercom, just the impression I get from Josephine Public!)
I'm wondering if she really won't try to get profesional help, or if she has been refused any more help to the point where it is soul-destroying to keep asking (yeh I might be projecting here). It's also possible physical symptoms are being dismissed as in her head/she's making a fuss. Not saying this is the case, but just to be aware it might be a possiblility. (I've been described as "refusing to seek help" before, by people who can't see any other reason for not receiving any help.)
she gets upset if I suggest that there are some things that only a professional can help her with. She thinks I'm rejecting her but I'm really not.
Part of me wants to scream in frustration at this. Not at you personally; just the whole situation, the way that society has sectioned off certain things and certain people's experiences and emotions and one just gets a tight-lipped non-engagement and "seek professional help". Actually you don't sound like you're doing this; I'm just explaining the general feelings that can be provoked as a human being who's struggling, when you feel you're somehow not fully human, with human problems, but an unwanted weirdo who must beg for scraps of support from the state/professionals without the time or resources to really help. I suppose what I'm trying to say is, support from friends is invaluable, and can never be replaced by professional help. Sure, some things (eg. therapy) need a professional, but stuff like having a cuppa and being able to have a little cry and say "this is what I'm feeling/remembering" doesn't. And most importantly, the "normal" stuff, eg. a walk in the park as Beer suggests. Actually this stuff is vital to just feeling like a normal person. So I guess make sure (or keep on making sure!) you're clear about the specific things a professional may be able to help with, rather than a more general "seek professional help".
Another couple of things here - IME I found it helpful to know I was seeing friends fairly regularly, it helped keep a certain stability rather than trying to manage and eventually crashing and needing obvious support from them. The other thing is about the help/support all being one way. As someone who needed a lot of support I found it very upsetting and frustrating. I'm happy to give as good as I get, in fact I want to. I'd happily do the washing up/paint the bathroom/whatever, to help out a busy mum friend whilst we have a natter and she's feeding the baby or just getting a moment to put her feet up or something! Knowing I am supported in return. I don't know if your friend would do this, but IME people just don't think to ask, as they get used to the dynamic being one person always helping the other.
Sorry this is long, doesn't cover everything but just some thoughts I wasnt sure would be mentioned by other posters.