If you read page 223 carefully, you see that the arguments to the function Chained-Hash-Delete are (T, x) and not (T, k). Now x is here the node to be deleted. That is, you have already found the node with key k to be deleted. So, yes, for a double linked list, the actual deletion is in constant time, but that does not include the time to search for the node to be deleted.

Abigail