PA28R Pilot reported engine compartment fire and gear up landing.

Narrative

The flight was planned as a short preparation flight for a cross country the next day. Essentially a few practice approaches and back down again. Bird nesting season is in full bloom and I completed a thorough (or so I thought) preflight removing the new full-cowl cover and nose plugs for keeping birds out. I proceeded with normal start; taxi and run-up procedures and then entered my flight plan into the Garmin 430. Executed a normal takeoff. At about 50-100 ft. altitude the Cockpit began to fill with smoke. It was too late to abort the takeoff so I opted for an immediate landing asking the aircraft ahead of me to extend his downwind so that I could make a very shortened pattern. I lowered the gear handle and opened the pilot's vent window which helped clear my watering eyes. My approach was high and I slipped to slow the aircraft and keep the smoke away from the left side. At about 50 ft. I stopped the slip and lowered full flaps. I began a normal flare and at this point the prop and flaps absorbed the landing. The aircraft came to a stop. I performed shutdown procedures and egress with my [dog] who was on his first demo flight. As I reached to turn off the master switch I noted that the gear safe lights were all out and the auto gear extension was in the proper position for an auto extension that did not happen. Better a gear up landing than a crash due to smoke inhalation.