IND CIC Controller working combined position with Ground Control described a questionable landing clearance event noting working position and being assigned CIC complicates the requirement to provide general supervision.

Narrative

I was working Ground Control combined to (CIC) Controller In Charge; this is very dangerous since I am in charge of the Tower; but must keep all my focus on my taxiing aircraft and cannot watch what the Local Controller is doing. I have brought this up with management several times to no avail. There was training on Local Control and Aircraft X was cleared for an approach to the airport; but never checked on the frequency; I tried to see if he mistakenly switched to my frequency got no answer. I then got very busy with taxiing aircraft when I heard the Local Controller clear Aircraft X to land. Aircraft X called me after landing and I taxied him to his ramp. Later; when I was not as busy; the Local Controller instructor told me Aircraft X never checked on the frequency to acknowledge the receipt of the clearance. Since we did clear Aircraft X to land; but never got a read back; not sure it this is a pilot deviation; but since I was to busy to inform the pilot he was never talked too. This is an on going problem here at IND Tower; very rarely does a Supervisor come to the Tower; and that means CIC is combined to a working air traffic position. I have asked and made suggestions that it be combined to clearance delivery since this is the least busy position in the Tower; or to have a mandatory time to have CIC stand alone position during our peak traffic times. In this situation I was not informed until after the pilot had departed his plane.