EHIC does not cover treatment in private hospitals, or costs such as mountain rescue, repatriation or lost property. You need Travel Insurance as well as your EHIC.
Despite UK government advice to keep your EHIC with your passport, MPI advises you to have it with you to avoid any problems if you need treatment. If you are skiing, keep your EHIC with your ski pass. You will need to produce it in the event of an injury, there will be no time to retrieve this from your hotel/chalet and an emergency replacement is no longer available.
To apply for an EHIC click this link – it takes you to the official NHS site where the EHIC is free.