Many colleges also have their own halls and not just "private" ones. Is that not the case where your DD will be studying?
If not then there are two other possibilities which may help. If the college is part of the University of London (and just about every college in London is with the exception of Imperial) then your DD can apply to stay at one of the intercollegiate halls:-
halls.london.ac.uk/
Prices start from around £220 per week, although there are a few cheaper options.
There is also a housing service that the University run that puts students in touch with accommodation providers here:-
housing.london.ac.uk/
and here is a list of the colleges etc whose students can use the service:-
housing.london.ac.uk/about-us/member-institutions
Birkbeck, University of London
Central School of Speech and Drama (CSSD)
City University London
Courtauld Institute of Art
Goldsmiths, University of London
King’s College London (KCL)
London Business School (LBS)
London School of Economics and Political Science (LSE)
London School of Hygiene and Tropical Medicine (LSHTM)
Royal Academy of Music (RAM)
Royal Holloway, University of London (RHUL)
The Royal Veterinary College (RVC)
The School of Advanced Study
The School of Oriental and African Studies (SOAS)
St George’s, University of London
University College London (UCL)
Queen Mary, University of London, does not subscribe to the ULHS service. QMUL students who want support with private accommodation should contact Residential Services and Support at QMUL.
External Colleges
Architectural Association - School of Architecture
Guildhall School of Music and Drama
London Contemporary Dance School
Pearson College
Ravensbourne
Royal College of Art
Sotheby's Institute of Art
University of the Arts, London (UAL) consisting of:
Camberwell College of Arts
Central Saint Martins (CSM)
Chelsea College of Arts
London College of Communication (LCC)
London College of Fashion (LCF)
Wimbledon College of Arts