The gravitational focus you are talking about is actually a minimum value, defined by parallel rays of light from a very distant star just skimming past the Sun as they are bent according to General Relativity.
The general formula for such lensing is that light is bent through an angle (in radians) of
$$\alpha = \frac{4 GM}{c^2 r},$$
where $M$ is the mass of the lens (assumed to be a point or spherically symmetric mass) and $r$ is the closest approach of a light ray to the lensing mass.
To work out where a ring of rays will be focused is just a bit of trigonometry.
$$ d_f \simeq \frac{r}{\alpha} = \frac{c^2 r^2}{4GM}$$
This focal distance is a minimum because it would be larger for a ring of rays that passed the lens with a larger value of $r$.
For the Sun as a lens you use $M=2\times 10^{30}$ kg and $r=6.9\times 10^{8}$ m, and calculate $d_f = 540$ au.
White dwarf stars have a similar mass (actually most are about 60% the mass of the Sun, but Sirius B is almost exactly a solar mass), but have radii about the size of the Earth - i.e. a hundred times less than the Sun.
This means that the value of $d_f$ is going to be about 10,000 times less than 540 au. You can use the formula above to calculate it for any combination of mass and radius.
To use the telescope, you place detectors at your chosen focus and observe the bright "Einstein ring" of a distant source that is exactly behind the lens. The magnification factor (the increase in the amount of light collected from the source) then is $4\alpha/\theta$, where $\theta$ is the angular size of the source without the lens.
For a white dwarf, the magnification at the minimum focus would be 100 times larger, because $\alpha$ is 100 times larger.
Note that the size of the image is modified by the ratio of the focal length to the source distance.
$$ x_i = x_o \frac{d_f}{d_o}$$
Thus the image of a distant object will be 10,000 times smaller than if using the Sun, which is much more convenient!
e.g. Observe an Earth-like planet at 10 ly at a focus of 630 au (= 0.01 ly) from the Sun. The image diameter will be 12.5 km. That's a lot of CCD detectors! Using a white dwarf at a focal length that is 10,000 times smaller gives an image just 1.25 m across.
All this assumes that the telescope is perfectly pointed with the source right behind the lens. Any relative motion has to be corrected or the image will move through the focal plane very quickly (like a planet viewed with high magnification through a normal telescope).
