When Axel arrived at the hospital, he was treated for severe hypothermia. At the same time, the doctors fought to contain the infection in his leg. Treatment became complex because the ice was clearly a result of Axel using his power.
Peter theorized that Axel had turned his cold in on himself, hoping it would do exactly what it was doing: slow the infection. But, unfortunately, his body seemed to be on autopilot, continuing to freeze to stop the disease while killing Axel at the same time.
Not only the long-term effects of the cold on his body, but the actual prolonged use of his powers was draining him. As a result, life-saving measures had to be taken more than once to keep him from dying during treatment.
Alice had been through a thorough exam after finding Axel and returning to the hospital. She was cleared of medical concerns but confined to her room with orders to stay in bed.