Talk:Dirac Delta Function
not sure that you really can define the energy property.
for example, pretty much every source defines the fourier transform of the dirac delta as 1, and not 0. in such a case, parseval would argue strongly against finite energy.
likewise, while impulse responses would have finite energy they would not be equal-energy. eg, e^-kt u(t) as an impulse response is often used. but the energy of this is based on k.
thus, while a dirac can practically be used as a way of representing "a sudden injection" of energy into a system, the actual energy amount would be dependent on the actual system.