For the record, the asker changed the question after I answered, specifying that not declaring an emergency is a condition of the question, so I will address this at the top without modifying the original part:
An attitude indicator is legally required for IFR flight. If equipment becomes inoperative during IFR, the pilot should inform ATC. I do not know that there is any specific legal text that addresses this particular situation you're describing with an ILS approach coupled with an AI failure, but it would be generally accepted by pilots that losing an attitude indicator is an emergency as opposed to losing other required IFR equipment such as the clock. So, I'm going to go with no, not without declaring an emergency.
In that situation depending on the circumstances, declaring a missed approach, an emergency informing ATC of the system failure, and requesting vectors to VMC conditions would be the best option.
In an emergency, you can deviate from rules to meet the needs of the emergency at your discretion, so I don't believe that technically you can't still shoot to minimums for the ILS, but it would be dangerous, especially if the pilot isn't highly proficient, and should be a last resort with no primary gyros.
If an approach needed to be shot, and there was no VMC within range, a 'Precision Approach Radar' approach would likely be the best option if one existed within range. This is can be a 'no-gyro' approach upon request.