jsa, p/t work and housing costs

A client is working part time earning £78 a week.  Her employer will not let her increase her hours and she is struggling to pay her mortgage.  She has two children aged 15 and 18 who she gets CTC for.

She wanted to know if she could give up her work and claim JSA and therefore get housing costs included in her applicable amount.  I was concerned that she would risk being sanctioned if she voluntarily gave up work and also believe that she would not necessarily need to give up her job to qualify for housing costs. 

I was looking at the post January 2009 rules in relation to housing costs and it appears to suggest that if she was entitled to jsa only after housing costs were included in her applicable amount then she would still qualify for housing costs after 13 weeks if she put a claim in now. 

Have I understood this correctly?  Is there any other advice she could be given?

How many hours a week?

Sorry, forgot to say that she is working less than 16 hours.