Air carrier flight crew encountered severe icing during climb along with failure of the anti-ice system.

Narrative

Severe Ice accumulation. We departed ZZZ airport enroute to ZZZ1 we were planned at FL220 due to moderate turbulence at higher altitudes; we were in solid IMC climbing through about 4;000. Upon leveling at FL220 we were advised of light ice in the area; we got an ice indicator message; cowl anti-ice was already on; and we turned on wing anti-ice. After about 3 min or so we still had an ice indicator and started to accumulate ice on the wipers. So; I decided to ask ATC for a higher altitude; they gave us FL260. As we started the climb the severity of ice accumulation immediately intensified to the point of Rams Horns on the wiper blades; and ice along the front windshield; at about FL230 we had an Anti-Ice Duct Caution followed by Left-Wing Anti-Ice and Right-Wing Anti-Ice Caution immediately followed by an Anti-Ice Duct Warning and a Master Warning. The FO (First Officer) was about to read the QRH for the caution when we got the warning. At this point; we were picking up what seemed to be severe Ice on the window and wipers. As the FO proceeded to follow the QRH for the warning it advised to leave icing conditions (which we were attempting to do) and turn the wing anti-ice off. The next step in the QRH is a question Is ice accumulation observed on the heated portion of the wing leading edge"? It requires a yes or a no to continue the procedure. It is impossible to see any portion of the leading edge of the wing from the flight deck of a CRJ 900.Climbing through FL230 I immediately applied max thrust as I watched our airspeed slowly bleed off; we were only able to obtain a VS of about 500 FPM. Our airspeed was down to about 270kts. I advised the FO to assume the wings were covered with ice based on the aircraft's performance. As the FO was reading and following the QRH she recycled the wing anti-ice system; I pulled up the anti-ice synoptic page and saw we had lost all wing anti-ice capabilities and they had not come back on after resetting; the cowl anti-ice was operational. As our climb rate continued to slow and we continued to lose airspeed I realized our situation was dire as the aircraft's performance told me it was completely covered in ice. At this point I realized we could not continue the climb as our airspeed was bleeding off; we had max power and were only getting 400-500 fpm. I decided to advise ATC and start an immediate descent and ice dispersal procedure; as the FO and I were discussing how to handle the descent we saw blue and broke out on top at about FL250. We were showing about 260kts stable airspeed; trim seemed ok although I had not turned off the autopilot yet. Now in the clear not picking up any more ice we had some time to evaluate the situation. We ran the QRH again and recycled the wing anti-ice system again which now came back on based on the anti-ice synoptic page. We had the flight attendants look at the wings; but they could not determine anything; so the FO went back to have a look; she thought she saw ice but could not tell. As we were evaluating what to do next our first priority was to accelerate to VMO to disperse any ice; which took us about 75-100 miles. We then climbed to FL300 to remain on top of the all-weather. The QRH does not give any guidance about reentering icing conditions or whether the anti-ice system will remain functional or not once it fails and the QRH procedure has been completed. We attempted to contact dispatch and maintenance through commercial radio but were unsuccessful with maintenance. We were advised there was ice from about FL280 down to 9;000 going into ZZZ1. We discussed diverting to an airport in the clear or trying to make ZZZ1. Dispatch advised us it was now clear below 10;000 going into ZZZ1. We decided ZZZ1 was our best bet; so we planned to ask ATC if we could remain at FL300 as long as possible and advised them our anti-ice system was malfunctioning and requested a rapid descent from FL300 to 10;000. ATC changed our arrival and allowed us to stayhigh then gave us the clearance to descend directly to 10;000; we broke out somewhere above 10;000 and remained in VMC into ZZZ1. We had mere minutes from the moment we started to accumulate severe ice and received the caution messages followed by the master warning to assess the situation run the QRH procedures and evaluate the aircraft performance. The QRH procedure certainly added stress as well as additional time necessary to make decisions. An immediate descent was an option I considered but only being at FL230 knowing we had terrain below us around 4;000 feet with IMC all the way down while a cold front had just passed through the area a climb was my first choice as the CRJ 900 is supposed to be capable of operating in known icing conditions. I had no idea that severe ice accumulation in such a short time frame would cause the anti-ice system to overheat and fail."