If the shaft is worn excessively, you could build it up with JB Weld, then sand it down for a tight fit, but it doesn't look that worn. To hold the inner race in place, and be able to easily remove the wheel, you need side compression on the inner race, either with springs washers or a heavy spring. Threading the end of the axle then using a nut to hold the inner race firmly would be the best, but not the easiest. You could also try a pin [or a weld spot] in the axle wheel stop plate or the axle, then notch the inner race with a small grinder to that protrusion so that it's indexed into place.