ARGC didn't used to treat single ladies , and one single lady has tried numerous times for an appt and never got one and that was recently! They don't have a sperm bank either.
LWC isn't the only clinic who treats single women in London there are loads of others and you can either use the clinic sperm or import it in via the ESB or USA - providing you meet the HFEA criteria-
Why not pop over to the single girls thread and the ladies there will be more than happy to share their experiences.
http://www.fertilityfriends.co.uk/forum/index.php?board=403.0
I have used the Bridge, CRM, Create and been to the LIster - all whom treat single women in London but I don't need donor sperm- I was also treated at ARGC but as a partnership not single woman! Other London clinics that I know of that also do include Homerton, CRGH (UCH) and I believe Barts and Guys may do.
Not sure how old you are etc but the Lister and CRGH (along with ARGC) are the most successful London IVF clinics- CARE in Nottingham also in top 5 and they treat single women!
Good Luck