I think you misunderstood your lawyer. You do NOT have to stay in an adulterous marriage.
He said adultery was not grounds for divorce: In other words, Illinois is probably a no-fault divorce state, which means you don't have to have a reason or someone to blame in order to get a divorce.
You can just cite that there has been an irretrievable breakdown or that there are irreconcilable differences. That alone is sufficient.
He was not saying that there's no basis at all for a divorce, only that adultery is not a basis in your state.
And the reference to the kids or money is meant to say that adultery will only be "useful" to you in a divorce if it was done in front of the kids (which would help you in a custody fight) or if she was buying diamond cuff links for this guy out of the marital pot (which would help you in a property settlement).
Make sense?
Check w/a lawyer in your state to confirm what I said. Most offer free consultations....