Legal Question in Landlord & Tenant Law in Illinois

Potentially. If the landlord is responsible for extermination, you need to give proper notice and if the landlord fails to remedy the situation you may have cause for "constructive eviction" (where conditions are so bad it's like you're being evicted) but that is tricky and you should have an attorney help you with that process. Have you contacted the local building department for an inspection - a finding of infestation may help your case. Especially if you let the landlord know you plan on doing that. But beware - asserting your rights can lead to unintended consequences - the landlord may attempt to evict even if you have a defense - retaliation, or uninhabitability.