How bright is Andromeda's center, in magnitudes per square arcminute or something similar?
The surface brightness number depends on a few things: how large an area of the image is averaged, how close to the galactic nucleus it is, and in what wavelength band.
Some papers give a brightness profile along a line bisecting the galaxy image.

Sandage, Becklin, and Neugebauer 1969 report, for a 7.6 arcsec diameter circle centered on the nucleus and measured in the $V$ band, a total magnitude of 11.3 and an average surface brightness of 15.5 mag/arcsec$^2$.
The same aperture 30 arcsec east or west of the nucleus was 1.5 magnitude fainter.

Kent 1983 images the central bulge in the $r$ band and finds a surface brightness of 14 mag/arcsec$^2$ at the nucleus and 16 mag/arcsec$^2$ 10 arcsec away along the major axis of the galaxy image.

Walterbos and Kennicutt 1987
measure a 60 arcsecond circle and find a surface brightness,
1 arcminute away from the nucleus along the major axis,
of 18.5 mag/arcsec$^2$ in the $B$ band.