Patricia, The profile has not been deleted. I can get to the page that indicates it is not visible because the person is living, which is exactly as it should be. However, on that page, it invites me to contact you, as the profile manager. As manager, you should be able to see the profile.
Your cousin can't just sign in - he has to join WikiTree or he can accept an invitation he will receive automatically if you enter his email address on his profile. That is probably the easiest way to do it - just edit his profile and enter his email address, then all he'll need to do is click the link in the message WikiTree will send him to accept the invitation and he will automatically become manager of his profile.