Answer: 5 kg

Let $m$ be the mother's weight, $b$ be the baby's weight and $n$ be the nurse's weight.

We have
$$
\begin{align}
m+b&=78\\
b+n&=69\\
m+n&=137.
\end{align}
$$


Adding all three equations together
$$(m+b)+(b+n)+(m+n)=78+69+137.$$ It follows that $2(m+b+n)=284$ and hence $m+b+n=142$

Since mother and nurse together weigh $137\;\mathrm{kg}$ the baby must weigh $5\;\mathrm{kg}$


Removing the nurse and mother
$n+m=137$, and$\ n+b+m+b = 78+69\\
\Rightarrow n+m+2b=147$
$\therefore 2b=10\Rightarrow b=5$