calculate the gcd of $a+b$ and $p^4$
The final equation is wrong. First, you need also $\,\color{#c00}{p\nmid k}.\,$ Therefore
$\qquad(a\!+\!b,\,p^4)\, = \,(kp\!+\!\ell p^2,\,p^4)\, =\, p\,(k\!+\!\ell p,\,p^3)\, =\, p,\,\ $ by $\,\ p\nmid k\!+\!\ell p\ $ (else $\,\color{#c00}{p\mid k})$