I wasn't able to do the nested sum in a single step, but breaking it into two slightly simpler nested sums might be valid. I also wasn't able to sum k from zero to infinity in the second summation without it timing out or saying that it could not understand my input, but summing k from zero to 20 looks like that is very very close to the infinite sum, even k from 0 to 6 gives almost the same result..

Sum[2^k 87^(2 k)/(622^(2 k) k!)*1, {k,0,Infinity}]+
Sum[2^k 87^(2 k)/(622^(2 k) k!)*-Exp[-215^2/622^2]*Sum[2^j*15^(2*j)/(622^(2*j) j!), {j,0,k}], {k,0,20}]

See if you can understand how I rearranged that and if you think you can justify the changes I made..