How can I check that the winner has the dlc ?

Add them on Steam and check from the storefront. It'll tell you if they already own it.

The only absolute method is to add the user on steam, add the dlc to your cart and choose to buy as gift and send directly through steam. You dont need to buy it but it will tell you which friends own it.

^
Or go to gift to Friend, it may say whether person has the DLC. Definitely works with games and some DLCs. Not sure if works with all DLCs.

EDIT: Sorry, my way applies to gifts. For keys, is the method immediately above.

To give dlc's is a pain ... I sent him an invite ... let's see what happens :D !