Pilot reported windshield heat failure and Pitot/Static heat failure while in moderate icing conditions. The pilot requested priority handling and descended to exit icing conditions. The pilot performed a precautionary landing at destination airport with unreliable altimeter and instruments; resulting in several altitude and heading deviations.

Narrative

Over the mountains enroute to ZZZ from ZZZ1 I entered moderate icing. The icing continued for the remainder of the flight. Prior to entering icing I turned on all my icing equipment including windshield anti-ice. The windshield anti-ice had failed. This resulted in me accumulating a significant amount of ice on both windshields. In addition the Captain's side attitude indicator became partially tumbled/unreliable when I entered moderate turbulence. The Attitude indicator bank was drifting about 10-15 degrees to the left and the pitch attitude was not consistent. I advised the Approach Controller that I had a windshield icing failure and will likely have to circle near the airport. While cleared to descend to 9;000 ft. I momentarily descended to 8;650 ft. due to the culmination of problems that I was troubleshooting at the time. ATC was vectoring me for the RNAV XXL however I would not be able to land because of the no forward visibility. The safety of flight was in question so I requested priority handling. The airport was VFR at the time; I was given direct ZZZ. Once out of the clouds I was cleared for a visual approach and circled east of the field at about 2;000 ft. AGL. The lower altitude was warm enough to thaw the window and I was able to regain forward visibility and conduct a safe landing on XXL.I became aware of the windshield Anti-ice failure when the ice started to accumulate on the window. I noticed the attitude indicator failure when my indication was not aligning with what my other instruments or co-pilot attitude indicator. I noticed the altitude deviation at 250-275 ft. low; ATC was quick to say 'Aircraft X 1;450 ft.; 9;000 ft.'.The windshield anti-ice had a mechanical failure resulting in the forward windshield becoming covered with a layer of ice. The attitude indicator had a mechanical failure as well and became unreliable. Trouble shooting combination of problems with moderate turbulence resulted in my attention split and got behind the airplane and leveled off 350 ft. low. For the attitude indicator unreliable I 'pulled to cage' that brought it close to level. However; it continued to drift and was unreliable. The windshield anti-ice failure prompted me to try and trouble shoot. I attempted the windshield anti-ice on both without any success. I searched for popped circuit breakers; made sure the window defrost was open. This was a significant issue and I advised ATC the nature of the problem and my intentions about 30 miles out from the airport. They gave me a heading that got me out of most of the icing. Shortly after I was cleared to descend to 9;000 ft. and momentarily went 350 ft. low. I recovered promptly and expeditiously. I took a second to focus solely on flying the airplane. 'aviate; navigate; communicate'. The ATIS was reporting VFR and I was confident I could circle over the airport visually and get the ice on the front windshields to thaw. I asked ATC to circle over the airport. At this point I entered the clouds and icing again. I [requested priority handling] because I was not confident I could safely land on the approach. Immediately after [requesting priority handling]; I was given direct ZZZ. I was able to enter VFR conditions and called airport insight. They cleared me for visual approach and to circle east of the airport. I did a few circles using visual reference until the majority of my windshield was clear of ice. I advised ATC and joined a left base to XXL. Landed with no issues.I personally done the Friday checks a few days prior in Aircraft X and checked the windshield anti-ice. It was working at that time. It seems to be a random mechanical failure which is hard to prevent. This also applies to the attitude indicator as well. My training and system knowledge gave me the tools to trouble shoot; adapt; and make a decision safely meet the needs of the situation. The 350 foot altitude deviation can be prevented by slowing down. I was doing too much at once and fell behind theairplane momentarily.