I'm trying to understand decidable languages. In particular, I would like to show that $$B = \lbrace \langle D \rangle \mid \exists k \geq 0 \,.\,\text{DFA $D$ accepts $a^k b^k$}\rangle.$$ I don't quite understand the process of proving these. I know that $a^kb^k$ is not regular, so then no DFA accepts it. I also know that $A_{DFA}$ (acceptance DFA) is decidable, I also know several other decidable languages like $E_{DFA}$ and $EQ_{DFA}$. How can I use these to prove that $B$ is decidable?
If no DFA accepts $a^kb^k$, doesn't that mean that $A_{DFA}$ will reject? So if $A_{DFA}$ rejects then shouldn't the decider for $B$ accept?