The problem with "cheap" storage will be that they can't afford it.
If they've got a limited budget for renting, so limited they can't find anything and have gone down (albeit they failed to stick to and been discharged from) the homeless route for a council house, they're probably looking for something to rent that's within the Housing Costs element of UC. There will be almost nothing and what there is will be shite in some way or other and fiercely fought after by hundreds of others in the same situation as well as those not on benefits who just want cheapest rent, meaning landlords can take their pick.
The only way they can get a better property would be by using some of the rest of their UC/wages to top up the rent, renting somewhere with less bedrooms than the number of bedrooms they need and have been allocated costs for, or getting a council house that's in a good state of repair. They have turned down the latter and probably won't accept either of the former.
If they had £100/month for a storage unit they could instead put that £100 towards their rental budget, but they either can't spare the money or won't on principle.
Now they're living in a hotel. Which is going to be costing them a lot more than the temporary housing the council gave them under the homelessness scheme.they were probably in a flat if they've got DC and felt comfortable to remain there, shown by turning down a permanent house for being "too small". Now by making themselves voluntarily homeless they've been kicked out of that accommodation.
Under the scheme, whatever temporary housing they give you, including B&B, the cost is paid by UC/HB however high it is, because it's not your choice to live there, the council has placed you there. Now they're voluntarily homeless they may only be receiving the standard amount of housing costs in benefits and having to pay the extra for the hotel themselves because they've caused this situation themselves. They could have been permanently housed somewhere the rent would not have exceeded the allocated housing costs, but they turned it down. The government is under no obligation to continue giving them extra funds to support their lifestyle choices (a hotel). Even if, because of the children, the council hasn't actually kicked them out onto the streets to fend for themselves. They were apparently broke before, they're even more broke now.
This is why giving them a deadline is pointless. They'll either come collect with an ultimatum or they won't. Giving them a deadline won't change that, it just delays the inevitable and lessens the chance they'll collect because it gives them time to decide not to bother. They don't need a month to sort out a storage unit, they can sort one out in an hour if they're going to do it. If they're going to beg space off another friend they can do that in a day of contacting everyone and are more likely to be successful with that if they can claim it's an emergency and their stuff has been been thrown out to the pavement/lawn. Give them a month and they'll either do absolutely nothing at all or they'll do whatever they could have done today, in a few hours on the last day.
The friendship is over, there's no saving it no matter what you do. Piss takers are only friends with people who allow them to do it. They may remain low grade friends with people with strong boundaries just in case they get to take the piss in the future, but the one thing they won't ever tolerate is people previously without sufficient boundaries suddenly getting some and putting their foot down with them.