January starts are hard but not impossible to find, funding and academic years just dont work easily to January starts. I suggest you call around all the Further Education Colleges in your area and ask. There will be some who offer level 1 courses for NEETS (young people who are not in education or training) that start in January.
Why is she in this situation? Your daughter needs to take a pro-active role in this if she is able. Your needs for money are undertandable but they will not motivate her to attend and if she does not, or has a poor history of attendance she may find it difficult to find somewhere that will take her.
I just googled "college courses starting in January 2016" and I can see a whole list of colleges from Westminster to Newham offering new year starts. Good luck.