If you don't have a copy of the relevant ordinance, here it is.
https://www.chicago.gov/content/dam/city/depts/bacp/Small%20Business%20Center/sharedhousingordinance...
The section regading buildings with 5 or more units is on p23. Item 4-14-060 (f). Not more than 6 units or 25%, whichever is less. One could apply for a Commissioner's Adjustment - see p25 item 4-14-100.
Basically, you have to write a letter outlining why you think you might be eligible for an adjustment, based on the considerations they outline in the ordinance.
The "climate" as regards short term and vacation rentals is not very friendly at the moment - even if you appeal, you should prepare for a denial. You should read section 4-14-90 which talks about all units in a building potentially coming under review, if there is a denial. You might want to find out if the other host is active, and whether others in your building would support you.