What to do in the case of an accident (discussion)

I'd like a bit of advice for an incident that happened 2 thursdays ago

I was cycling down west norwood high street , when i hit a car that had pulled out quickly from a side road, going over the bonnet and landing on my face - the woman was apologetic and told me that the person turning left onto the side road blocking her view had told her it was safe to come out. luckily alot of people actually seemed to care so when i was taken into the ambulance i was handed the details of a witness for me and somebody had already contacted the police.

I was advised to persue a claim for my injuries but I have no idea how to go about it and cannot afford myself a solicitor - so i went ahead and called National Accident Helpline

I was meant to have a meeting today at my house at 8, but the person I was meant to meet didn't arrive. after me calling at 10 o'clock the person told me she had no details of the appointment or my case and , after another hour, she called back and we organised to meet tomorrow. 2 minutes later she calls me, not having a clue that she'd called me, saying I had an appointment at 12 today, clearly having no idea she had spoken to me just a few minutes ago.

Now this sort of imcompetency worries me (first impressions are pretty terrible at the moment) if this person is handling my case, so any recommendations of what I should do? I would have thought the case is pretty clear cut. I can't afford to do anything but no win no fee, are there small (competent) firms that do this or can somebody reccomend a firm they've used before?
Just Google for a London solicitors and find someone who deals with litigation. If you have a witness and it wasn't your fault, most solicitors will want to take your case as they get to charge the person you're suing for their fees (i.e. they are given an easy meal).

Now for my problem... 2 weeks ago I was cycling along a road (going pretty fast, maybe 25mph) and a car pulled out from a side road and I hit straight into her bonnet. She basically admitted liability by saying that she wanted to pay for my bike (the steel frame was totalled) by cheque instead of going through insurance. I said that was not acceptable as I had hurt my knee and if it didn't heal properly I would want compensation. I have just found out yesterday that I have cartilage damage, so it's not gonna heal properly :'( Two questions - I have no witnesses or CCTV other than me, her and her passenger. Although a pub full of people laughed as the accident happened, none came over to help and I didn't feel like approaching them to ask for their details (chav pub). Is there likely to be any problem with claiming against her? She is totally at fault and also admitted liability, however I understand insurance companies might try to wrangle their way out of this by the fact there are no independent witnesses. Also, what timeframes are associated with this sort of claim? My solicitor said "I can't see how they could deny liability" and it "could take several months" but I am not particularly satisfied with his vagueness - would be great to have a second opinion.

Cheers