The Michelson interferometer compares the lengths of two light paths.
One source produces two light beams which appear to come from two "coherent" (virtual) sources which traverse two paths and then superpose to produce an interference pattern consisting regions of different intensities - light and dark fringes.
Amongst other factors the separation of the fringes depends on the wavelength of the light.
The shape of the fringes depends on the orientation of the mirrors which if exactly at right angles to one another produces circular fringes and if at other angles to one another can produce straight line fringes.
White light contains a continuous range of wavelengths each of which produce an interference pattern with a different fringe separation.
It is only when the path lengths of the two beams which form the interference pattern are equal that constructive interference occurs at the same region for all wavelengths and that is often called the zero order fringe and is shown below in the middle of the diagram.
What you see beyond the zero fringe is the fringes pattern of different wavelengths overlapping one another.
So for the first bright fringe beyond the centre (first order fringe) it is blue (short wavelength) on the "inside" and violet" (longer wavelength) on the outside because the blue light has the smallest fringe separation and the violet light the largest fringe separation.
As one moves away from the centre you could have a bright fringe due to one wavelength occurring at a position where there is a dark fringe of another wavelength.
Thus moving out from the centre the fringe patterns due to each of the wavelengths which make up white light overlap so much that no further fringes are discernable.
With while light one might see a central white fringe and a few coloured fringes and then a uniform illumination.
To measure wavelength using the interferometer changes one of the path lengths by moving a mirror a measured distance and counting the fringes which cross a graticule in the field of view and each complete fringe movement corresponding to the mirror being moved half a wavelength of the light.
To get a reasonable estimate of wavelength one might measure the distance the mirror moves when $50$ fringes traversed the field of view.
The problem with trying to measure a wavelength of the light which makes up white light is that as soon as a mirror is moved only a little from a position when the light path lengths are equal the white light fringe disappears, coloured fringes replace it and then as the mirror moves further the fringes disappear.
One could use narrow width bandpass filters to measure the wavelengths of some of the components of white light.
One advantage of using a white light source is that it can be used to set up the interferometer with equal path length to within a fraction of a wavelength as only then will a zero order white fringe will be visible.