Hi rzellicks and welcome to FlyerTalk!

If your DOH-MLE flight is timed under 5 hours and if that flight departs no later than 6am the following morning after you arrive at Doha, yes, you would be able to access the Al Mourjan Lounge. Although, you might be asked for proof of your Business Class travel from Chicago to Doha, thus don't throw away your boarding pass for that leg before entering the lounge.

If your DOH-MLE flight is timed over 5 hours, you would only be able to access Al Mourjan by paying the respective entry fee ( $96 for each 6 hours ).