How can the landlord increase the rent by 30%? Have they done a lot of renovation work in the last month or two???

Regardless, you're responsible for the rent up to the end of the notice period. That obligation is relieved if you make genuine efforts to introduce a Nachmieter (even if the landlord rejects your proposed tenants). However, it is unreasonable to expect you to find a Nachmieter at 130% of the current rent (if that is even legally permissible).

As curley said, if you have an email that states explicitly that you do not need to find a Nachmieter and are off the hook as of 31 January -- and not "that they probably release us from contract obligations" -- then I'd say you're in the clear. No need to pay past January, and you have documentary proof.

ETA: beaten to the "probably" by Edwin.