The mouse isn't killed it is captured and the page is moved forward 1. You loose all your non starting items and can come back when all enemies on the tile are killed when it is next your turn if there are still non captured mice.
Tilda is a healer. When you start the game, you get to choose an ability for each mouse. One of the abilities for the healer class is to spend two cheese and heal wounded mice (non-poison wounds). There are proximity requirements. Check out the ability card.
There is another ability that enables the healing of poison wounds.