I would be swayed by the fact you are obviously very unhappy in your current role, if you are currently dreading going into work every day them it is certainly time for you to get out and find something new.
Obviously working on a temporoary contract has its potential pitfalls - worst case scenario you end up jobless after a few months but best case scenario is that you could at some point be considered for a permanent role when a vacancy arises. I know my employers will always try to give a permanent position to any agency staff who have proved their worth when one arises.
Being in a position to financially support yourself for a few months if it all falls down is essential but it looks like you have the covered. You can always continue to search for a suitable permanent position and if the new company is a well respected one then that looks good on your CV.
Don't forget to check if the 'employee care' benefits also apply to temporary staff.
I would also ask the new company if they would be prepared to wait until you work your 4 weeks notice - if they are then it shows they are keen to have you work for them as all companies know that most people are required to give a months notice. If not then no matter, as Theresa said someone not working out their notice is not something that a company is likely to pursue, though it can of course affect your references.