Infinite number of intermediate fields between K(u,v) and K
Solution 1:
You're on the right lines. Note that since $a \neq b$, then $a - b$ is invertible in $K(u+av)$, so $(a-b)v \in K(u + av) \implies v \in K(u + av)$. But that implies $u \in K(u + av)$ too, i.e. $K(u + av) = K(u, v)$.
Now you need to argue that this cannot happen. What is the degree of the field extension $[K(u + av):K]$?