OP I am a landlord and if you were my tenant I would be horrified, embarrassed and ashamed.
Feelings aside, protocol around how to respond in this situation should be clearly outlined in your tenancy agreement, including how to proceed and allocate costs when temporary accommodation is necessary.
The issue that you may come up against is whether the landlords service charge or insurance policies cover infestations, meaning that the landlord may have to bear the cost of the treatment and also any time you're staying away himself. I haven't checked my LL insurance but I do know it's not covered in our home policies as I checked. S/he will be much less accommodating knowing they have to pay for it all than if it were covered. Even if treatment is covered, it's unlikely that costs of alternative accommodation would be.
Personally I would absolutely release you from the contract and deal with the problem properly. It's very unlikely you're the only property in the building with the issue though so it could be quite major. The cost of resolving this issue is an unknown at the moment and your landlord may find it's cheaper to have the place unoccupied than to cover your staying elsewhere etc.
If you do leave though, the other issue then becomes sunk costs such as tenancy agreement fees, inventories, pre-tenancy cleans (and you might find yourself paying for post tenancy also) background checks, TDS fees etc. It's not that straightforward just to let someone go.
Personally I look to my letting agent for contract advice (but I manage our property myself otherwise), you need to be escalating this as far as possible. I find it impossible to believe that no signs of this were picked up in the inventory inspection and if the agent has rented out the property before they could also have known about it. There is a chance you could push it back on them too, did you inspect the property on arrival and did you see anything (even with hindsight) that looked suspicious?
Have a set of clear what if questions ready to email tomorrow morning and tell them you want an answer by cob tomorrow. Good luck.