You can reduce CLIQUE ($G$ has a clique of size $k$) to KITE: given $G=(V,E)$ and $k$, just build in polynomial time a new graph $G'$ in this way: for each node $v_i$ add a tail of $k$ new nodes.
- check trivial case $|V| < 3$
- recursively remove degree 1 nodes from $G$ (existing tails)
- for each remaining degree >1 node $v_i$ add a tail of $k$ new nodes
If $G'$ has a kite of size $2k$ then the $G$ has a clique of size $k$ (the kite without the tail). Removed nodes cannot be part of a clique in G, so $G'$ contains all cliques of $G$; addedAdded nodes cannot introduce new cliques on G′, so $G$ contains allexactly the same cliques of $G'$.