Pilot reported inoperative vacuum system and failed to follow proper MEL procedure prior to flight.

Narrative

I was flying our club's Aircraft X for the first time since we had Maintenance performed on the landing gear shock system. I started the aircraft normally and after obtaining taxi clearance from ZZZ Ground Control; taxied toward Runway XX on Taxiway 1. During taxi; I adjusted the heading indicator; and at the runup area; I performed the rest of the takeoff checklist. I must have been distracted; because I didn't complete all items on the checklist; including checking the vacuum system. I attribute this to not having flown for over a month. I contacted Tower for a takeoff clearance; and when received; taxied onto the runway. It was then that I discovered the heading indicator was way off; by about 180 degrees. The 'Instrument Air' gauge read zero; indicating the vacuum system was inoperative. Since I had already received takeoff clearance; I made a rapid assessment of the situation. I was in good VMC; flying under VFR; and the vacuum system only powered two instruments; the attitude indicator and the heading indicator; neither of which are required for VFR flight. I had a functioning magnetic compass; so I elected to takeoff and simply ignore the indications on those two instruments.I flew direct to ZZZ1 without further incident; completed two touch-and-go landings followed by a full stop landing. I then taxied to the fuel pump to refuel. I then prepared for takeoff again; after which I performed four more touch-and-goes and then proceeded direct to ZZZ. When I arrived at ZZZ; the control tower had closed; and ZZZ area procedures for closed tower operations were in effect; mandating the use of CTAF for position reporting. There were a number of aircraft on final approach for Runway XX. I announced my position West of ZZZ; and overflew the airport to enter a left downwind leg. When I eventually turned final approach; the wind had picked up and had a gust factor. When I got close to the runway; I found myself getting behind the airplane; and over correcting for changes to wind direction and speed. I ultimately decided to go around and try again. As I was climbing out from the go-around; I realized that multiple approaches were not authorized in the area without a SFRA flight plan after the tower had closed; so I attempted to contact ZZZ Approach to advise them I was reentering the pattern for landing. Although I could hear the controller talking to other aircraft; I was never called; so I switched back to CTAF; announced my position; and landed normally. At no time was the lack of an AI or HI a factor in these two flights. I had full situational awareness; using a combination of pilotage and data from the GPS on board; as well as the moving map display on my tablet computer.On the ground; I researched FAR 91.213; Inoperative Instruments and Equipment; and while VFR flight is authorized in this make and model of airplane without an operating vacuum system; or attitude indicator and heading indicator; I should have aborted my initial takeoff from ZZZ and complied with paragraph (d)(3) of that regulation; which requires the non-working instruments to be removed from the aircraft or placarded. I simply consciously ignored their indications; and that was not a factor during the flight. Following the flight i covered the inactive instruments and labeled them 'inop'; and added another temporary placard that said 'vac sys; attitude and heading indicators inop'. I also added an entry to that effect in the aircraft logbook in accordance with FAR 43.9. Subsequent to this flight; the aircraft was flown to a maintenance facility and the vacuum pump replaced; restoring the aircraft to full airworthiness and the temporary placards removed. The go-around at ZZZ was based on my assessment of a less than stabilized approach; and I felt it was much safer to go around; irrespective of the area limitations on repeated approaches when the Control Tower is closed.