= Solution
Write $N=N_G(D)$, $C=C_G(D)$, and use the commutative transfer <ideals>
$$
I_D=\operatorname{Tr}_D^G((kG)^D)\subseteq Z(kG),\qquad
J_D=\operatorname{Tr}_D^N(kC)\subseteq(kC)^N.
$$
The target fixed <algebra> is commutative, because $C\le N$ makes every $N$-fixed element of $kC$ central in $kC$.
The <Brauer morphism and relative trace> identity is
$$
\operatorname{Br}_D(\operatorname{Tr}_D^G a)=\operatorname{Tr}_D^N(\operatorname{Br}_D a).
$$
Indeed let $D$ act on $G/D$ by left multiplication in the trace sum. Its fixed cosets are exactly $gD$ with $g\in N$; every other orbit has size divisible by $p$. After Brauer projection the contributions on each such nonfixed orbit are equal, so they cancel. Fixed cosets give precisely the right-hand side. Since the Brauer projection of $(kG)^D$ is all of $kC$, this proves a surjection \b[$I_D\twoheadrightarrow J_D$].
We use the <trace criterion for defect groups>: a block <idempotent> $e$ belongs to $I_D$ exactly when its defect <group> is conjugate into $D$; among these, $\operatorname{Br}_D(e)\ne0$ exactly when its defect <group> is $D$ up to conjugacy. The criterion can be read from the class-sum basis: if $E=C_D(x)$ then
$$
\operatorname{Tr}_D^G(\operatorname{Tr}_E^D x)
=\operatorname{Tr}_E^G x=[C_G(x):E]\,[x^G].
$$
This coefficient is nonzero exactly when $E$ is a Sylow $p$-subgroup of $C_G(x)$. Thus $I_D$ is spanned by classes whose class defect is conjugate into $D$, and the kernel of its Brauer projection is the sum of the corresponding proper-subgroup transfer <ideals>. The block version follows by refinement of the <central idempotents> in these <ideals>.
For clarity, the refinement fact needed here is <primitive idempotents under a surjection of commutative Artinian ideals>: a surjection between <ideals> of commutative Artinian <algebras> bijects the <primitive idempotents> in the <module source> whose images are nonzero with the <primitive idempotents> in the target. Decompose the <algebras> into local factors. An <ideal> in a local Artinian factor either contains the identity and is the whole factor, or lies in its nilpotent maximal <ideal> and has no nonzero <idempotents>. A surviving local factor has a local quotient, so its identity remains primitive. This proves the stated refinement fact without imposing nilpotence on the entire kernel.
The <primitive idempotents> in $I_D$ are the block <idempotents> with defect conjugate into $D$, and the surviving ones have defect $D$. Applying refinement to the surjection above gives
$$
\boxed{e\longmapsto\operatorname{Br}_D(e):
\{\text{blocks of }G\text{ with defect }D\}\ \longleftrightarrow\
\{\text{primitive idempotents of }J_D\}.}
$$
This is the <Brauer morphism on a defect transfer ideal>; the target is the transfer <ideal> $J_D$, not the entire fixed <center of an associative algebra>.
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