Kademlia is the de facto standard searching algorithm for P2P (peer-to-peer) networks on the Internet. In our earlier work, we introduced two slightly different models for Kademlia and studied how many steps it takes to search for a target node by using Kademlia's searching algorithm. The first model, in which nodes of the network are labelled with deterministic IDs, had been discussed in that paper. The second one, in which nodes are labelled with random IDs, which we call the Random ID Model, was only briefly mentioned. Refined results with detailed proofs for this model are given in this paper. Our analysis shows that with high probability it takes about $c \log n$ steps to locate any node, where $n$ is the total number of nodes in the network and $c$ is a constant that does not depend on $n$.