Need direction: Tenant breaking the lease - bed bugs

[Marcin Czaicki]

Hello everyone. 

I would like to seek your opinion on my recent development with one of my tenants. They signed a lease in November for 12 months. In early January they advised that they have bed bugs which supposedly were confirmed by the local exterminator chain. I had an exterminator of my choice (licensed) to perform the treatment but after 1 month they claimed that the bed bugs are back. I involved the village and they sent out an inspector to investigate but the tenant never let the inspector in and just last weekend sent me a notice that they will be moving out at the end of April. This is a breach of the lease and as of today they didn't pay their April rent. 

I will be delivering a 5 day notice to them today and I do have their deposit which I was planning on using for the bedbug treatment but now it looks like I'm also going to be out of April's rent. 

Should I go through formal eviction with them even when they are moving out at the end of the month or should I just wait until they move out to start legal process for damages/costs? 

My lease does have clauses that define Abandonment and that tenant will be responsible for any and all damages/losses incurred by me due to their abandonment. This is a Chicago suburbs situation (Kane Co.)

Appreciate any and all responses.