Real Estate Law
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You have to give the landlord a chance to cure but if they fail you have the right to terminate the lease. They are required to provide a safe and habitable premises.
You will have to sue for breach of warranty in order to get the judge to allow you to break the lease. You can withhold rent for your extermination costs. The Landlord will sue you but then you would counterclaim that they breached the warranty of habitability. This is a good resource for you...http://www.ag.ny.gov/sites/default/files/pdfs/publications/Tenant_Rights_2011.pdf
That is a common frustration. Even enforcing our rights can be expensive. I would say withhold rent until the issue is addressed completely.
If you could rate the answer I would greatly appreciate it.
Good luck to you. One more tip would be to take pictures of any rodent activity. Those pics will be critical should either party seek court intervention.