Help with combinatorial proof of binomial identity: $\sum\limits_{k=1}^nk^2{n\choose k}^2 = n^2{2n-2\choose n-1}$

$$\sum_{k=1}^nk^2\binom{n}k^2=n^2\binom{2n-2}{n-1}\;.\tag{1}$$

You have $n$ men and $n$ women. You want to choose a team of $n+1$ people, and on this team you will designate a man and a woman as co-captains. You can choose the co-captains in $n^2$ ways, and you can then choose the remainder of the team in $\binom{2n-2}{n-1}$ ways, so the righthand side of $(1)$ counts the possible teams.

Now let’s count the teams with exactly $k$ women, for $k=1,\dots,n$. There are $\binom{n}k$ ways to choose the women, and $k$ ways to designate one the female co-captain. There are then $n$ ways to pick the male co-captain and $\binom{n-1}{n-k}$ ways to pick the remaining men. And

$$kn\binom{n}k\binom{n-1}{n-k}=kn\binom{n}k\binom{n-1}{k-1}=k^2\binom{n}k^2\;,$$

so the lefthand side counts the same thing according to the number of women on the team.