Let $\mathbb{F}_{2^m}$ be a finite field of cardinality $2^m$, $λ$ and $k$ be integers satisfying $λ,k\geq 2$ and denote $R=\mathbb{F}_{2^m}[u]/\langle u^{2λ}\rangle$. Let $δ,α\in \mathbb{F}_{2^m}^{\times}$. For any odd positive integer $n$, we give an explicit representation and enumeration for all distinct $(δ+αu^2)$-constacyclic codes over $R$ of length $2^kn$, and provide a clear formula to count the number of all these codes. As a corollary, we conclude that every $(δ+αu^2)$-constacyclic code over $R$ of length $2^kn$ is an ideal generated by at most $2$ polynomials in the residue class ring $R[x]/\langle x^{2^kn}-(δ+αu^2)\rangle$.