They are not bound. At white dwarf densities, a typical electron kinetic energy (0.6-0.8 times the Fermi Energy) is a few hundred to a few thousand keV. This is far too much for them to be bound to nuclei.
Details:
The Fermi momentum is given by
$$p_F = \left(\frac{3h^3 n}{8\pi}\right)^{1/3}\ , $$
where $n$ is the fermion number density.
The kinetic energy associated with the Fermi momentum is then
$$K = (p_F^2c^2 + m^2c^4)^{1/2} - mc^2\ , $$
where $m$ is the fermion rest mass.
This is the maximum kinetic energy of the electrons. The average kinetic energy is somewhere between 0.6 (non-relativistic limit, if $p_F \ll mc$) and 0.8 (relativistic limit, if $p_F \gg mc$) times the maximum.
If we take a typical white dwarf, made of carbon nuclei plus six electrons for each nucleus, the central densities reach $10^{10}$ kg/m$^3$. The electron number density is $n_e =\rho/2m_u = 3\times 10^{36}$ m$^{-3}$, where $m_u$ is an atomic mass unit and the 2 arises because there are two mass units for every electron in the gas.
Using the formulae above we get a maximum electron kinetic energy of 508 keV; the electrons are becoming relativistic, so the average electron kinetic energy is about 350 keV. This can be compared with the ionisation energy of the inner electron on a carbon nucleus of 490 eV.
More massive white dwarfs are denser, the Fermi momenta and average energies are larger.
If enough mass were added to the white dwarf, the maximum electron energy reaches about 14 MeV and it becomes possible for protons in the carbon nuclei to capture electrons to form neutrons (see https://physics.stackexchange.com/a/387013/43351). This process removes free electrons and hence the source of pressure and leads to the detonation of the star in a thermonuclear type Ia supernova.
