Simplify sum of Kronecker delta

Why not work with symbolic tensors instead? For example, your sum can be represented as:

s = TensorContract[
    TensorProduct[
        IdentityMatrix[j],
        IdentityMatrix[k],
        A
    ],
    {{2, 5}, {4, 6}}
];

The built-in function TensorReduce is not quite able to simplify this, but you can install my TensorSimplify paclet to enable simplification. Install with:

PacletInstall[
    "TensorSimplify", 
    "Site" -> "http://raw.githubusercontent.com/carlwoll/TensorSimplify/master"
]

Once installed, load the package with:

<<TensorSimplify`

Finally, let's try TensorSimplify on your example:

TensorSimplify[s, Assumptions -> A ∈ Matrices[{j, k}]]

A

One approach is to use DiscreteDelta instead of KroneckerDelta. With this substitution, the setup is:

$Assumptions = Element[a | b, Integers] && 1 <= a <= J && 1 <= b <= K;
expr = Sum[DiscreteDelta[a - j]*DiscreteDelta[b - k]*A[j, k], {j, 1, J}, {k, 1, K}]

A[Ceiling[a], Ceiling[b]]

Amazingly, it isn't even necessary to invoke Simplify, though I suppose it might in more complicated situations. To see that this substitution of DiscreteDelta for KroneckerDelta is correct, observe that

KroneckerDelta[a, j] == DiscreteDelta[a - j] // FullSimplify
True