expr = Sum[
   Log[Sum[A[j, k, l]*B[j, k, l], {l, 1, L}]], {j, 1, J}, {k, 1, K}];
expr = Simplify[D[expr, A[a, b, c]]]

expr /. Sum[
    y_ KroneckerDelta[s_, r_], {s_, 1, p_}] :> (y /. s -> r) /.
 Sum[y_ KroneckerDelta[s_, r_] KroneckerDelta[s1_, r1_], {s_, 1,
    p_}, {s1_, 1, p1_}] :> (y /. s -> r /. s1 -> r1)

expr = expr /. Sum[y_ KroneckerDelta[r_, s_], {s_, 1, p_}, z__] :>
    Sum[(y /. s -> r), z] /.
        Sum[y_ KroneckerDelta[r_, s_], {s_, 1, p_}] :>
            (y /. s -> r)