EMB-170 Captain reported during cruise; they got a bleed 2 fail indication while in icing conditions. They requested a lower altitude and while descending; the cabin pressure began to rise.

Narrative

Flight XXXX began at ZZZ Gate XX; with dispatch release including Bleed 1 inoperative. Passing 10;000 feet; ATC issued maintain 300 kts. or greater; later adjusted to 280 or better; climb & maintain 19;000 feet (top filed altitude). At approximately 16;000 feet; aircraft encountered icing. Shortly after leveling at 19;000 ft; still in icing conditions; EICAS displayed BLEED 2 FAIL. We initially requested lower (15;000 ft); but after noting cabin pressure rising; we [requested priority handling] and requested 10;000 ft. We pushed FLCH; speed 300 kts.; with half speed brakes deployed as cabin pressure was rising; but not at a rapid rate. We started the APU while in the descent to begin to re-pressurize the cabin. While in the descent A-I Wing Fail EICAS also displayed. When leveling at 10;000 ft; we were out of icing and at a safe altitude so we ran both QRH procedures. Bleed 2 reset successfully; but A-I Wing Fail EICAS did not extinguish. ZZZ Center asked if we were still a [priority] aircraft; and we said no. We continued the flight at 10;000 ft with the APU running and asked for direct to the final approach fix for Runway XXR. We opted not to make a call to the flight attendants (FA's) or the passengers as we were no longer in an emergency situation and we were only about 20 minutes away from landing. Landing and taxi in was uneventful.