Legal Question in Landlord & Tenant Law in Illinois

If I understand your question, you have a disabled relative and rented a 2-bedroom apartment to a couple to (a) care for the disabled person and (b) take care of the apartment, in return for which you waived cash rent. Since non-payment of rent does not apply here, your right to evict would be based upon their failure to comply with a leasehold requirement, being taking care of the disabled person. Normally this requires a 10-day notice (failure to comply with a leasehold requirement) after which, if they do not move voluntarily, you would have to sue to evict and they could defend on the basis that they did care for the person. In the interim you can certainly hire someone to care for the disabled person but whether or not you can claim the costs in your eviction case would require a complete review of the situation including any written lease you had with the tenant caregivers, and it is possible that if they extend the eviction process you may be able to get "use and occupancy" (as you say an "approximate" rent) but there is no way to enforce it during the eviction process and you might have to track them down for the amount awarded after the lawsuit is over. Have an attorney help you with this process.