She can contact the local council to make a homelessness application.
She is homeless, eligible and in priority need as she has a dependent child.
Therefore the council will have a duty to accommodate her in emergency accommodation. (section 188 Housing Act 1996) or consider her homeless at home at yours if that is preferable to emergency accommodation which can be crap)
There will also be a duty to relieve her homelessness as well.
This could be paying for a deposit or first months rent to secure a place in the private sector that is available for at least 6 months. (although please note the council can discharge the main housing duty at this point,
And as she was experiencing domestic abuse - she was already legally homeless even though she had a roof over her head so they cannot say that she has made herself intentionally homeless by leaving (they should not even be looking at intentionality at this stage anyway)
They will then look at intentionality and local connection.
Has she done or not done something to cause her homelessness (no she hasn't)
Has she lived in the area for 6 out of the last 12 months - does her child go to school in the area etc?
Once the council are satisfied that she "passes" all 5 hurdles they will have a duty to accommodate her in permanent housing. (s193)
This could be social housing or it could be a private let that is available for at least 12 months.
This could be out of area but she would have a right to challenge the suitability within 21 days of the offer.
If the council turn her away- (if they say that she is not homeless because she is living with you for instance)
then that is unlawful and the council can be challenged.
If she is on IS, ESA or JSA or has a low income she will be entitled to legal aid who will be able to assign a caseworker to challenge the council on her behalf. Google Civil Legal Advice for the number.
She can also apply to go on the waiting list. Most councils have priority banding for those considered homeless so she should get preference for bidding as well as the council doing their bit to house her.
It can be a bumpy road but the council definitely do have a duty (quite a few duties actually) Don't be put off by this. They can be challenged if they are not doing what they should be doing. (CLA advice above)
I appreciate this seems like long winded advice but if she knows her legal rights she is less likely to be fobbed off by the council.