Help with Help with Housing Costs.

Hi, its late in the afternoon, so I am posting to see if I am correct in my thinking….

My client (full time student with child) was on Income Support, but left the country for 5 weeks, so her entitlement stopped in August and she lost her transitional protection to IS.
She then claimed JSA, but of course has been turned down because she is a full time student. She applied again on advice of JC+ (!!) trying to argue that even though she was full-time, she didnt attend class (phD student) and was available for work. She also put in a claim for housing costs, which (she says) has been put “on hold” whilst they are deciding the outcome of the second application.
This is the point where she visited me yesterday.

I am trying to convince her to register as part-time, so she can claim JSA and help with housing costs.
She has never applied for housing costs before because her estranged husband was paying the mortgage. She was on Income support for 1.5 years when it stopped.
When I look up the linking rules for reducing the 13 week waiting period, I come to the conclusion that she will be able to make a claim for housing costs, but she will still have to serve the 13 week waiting period.

Am I correct in my thinking, or barking up the wrong tree. Any thoughts welcome :)