How can the nucleus of an atom be in an excited state?

First you say

It's easy to visualise and comprehend the excited states of electrons, because they exist on discrete energy levels that orbit the nucleus

By way of preparation, I'll note that in introductory course work you never attempt to handle the multi-electron atom in detail. The reason is the complexity of the problem: the inter-electron effects (screening and so on) mean that it is not simple to describe the levels of a non-hydrogen-like atom. The complex spectra of higher Z atoms attest to this.

Later you say

[nuclei] don't exist on energy levels that they can transfer between

but the best models of the nucleus that we have (shell models) do have nucleons occupying discrete orbital states in the combined field of the all the other nucleons (and the mesons that act as the carriers of the "long-range" effective strong force).

This problem is still harder than that of the non-hydrogen-like atoms because there is no heavy, highly-charged nucleus to set the basic landscape on which the players dance, but it is computationally tractable in some cases.

See my answer to "What is an intuitive picture of the motion of nucleons?" for some experimental data exhibiting (in energy space) the shell structure of the protons in the carbon nucleus. In that image you will, however, notice the very large degree of overlap between the s- and p-shell distributions. That is different than what you see in atomic orbitals because the size of the nucleons is comparable to the range of the nuclear strong force.

It is easy to think of the nucleus as a simple ball because it is so small (about 100,000 times smaller than an atom), but even at this level there is structure. While a different force mediates the interaction among nucleons (the parts of a nucleus), it is analogous to the interactions between nuclei and electrons that gives rise to the structure of atoms. These structures are a consequence of the quantum mechanical rules governing the interactions between particles and fields.

The nucleus of an atom is bound together through the strong nuclear force. This is one of the four fundamental forces, of which electromagnetism is a member. The many states of an given atom are governed by electromagnetic interactions between the electrons and the nucleus of an atom, with a ground state that represents the lowest possible energy configuration of the system, and excited states that are also allowable, but with higher energy values.

Similarly, there are many nuclear states for a given configuration of nuclei. Although mediated through a different fundamental force, there is still a ground state representing the lowest energy configuration of a particular collection of neutrons and protons, and there are many possible excited states as well. These excited nuclear states follow essentially the same rules that excited atomic states, except that the form of the potential term when writing down the Hamiltonian is different.