Deriving support in bidding strategy
I agree with the comments that suggest that the question is almost indecipherable. However, it should be easy, given the cdf, to derive the bounds. Why? Well we know for a cdf $F$, we must have $F(\underline{S}) = 0$ and $F(\overline{S})=1$. We can use the latter to work backwards from the upper bound i.e.

We guess that $F(s)$ is of the form

$$F(s) = k\bigg(\frac{S-v_l}{v_h-S}\bigg)$$

We must have $F(\overline{S}) = 1$ and using your solution, we have

$$\begin{split}F(\overline{S}) = F\bigg(\frac{7v_h+3v_l}{10}\bigg) = k\bigg(\frac{\big(\frac{7v_h+3v_l}{10}\big)-v_l}{v_h-\big(\frac{7v_h+3v_l}{10}\big)}\bigg) &amp;= 1\\
k\bigg(\frac{7v_h+3v_l}{10}\bigg)-kv_l &amp;= v_h-\bigg(\frac{7v_h+3v_l}{10}\bigg)\\
\frac{7kv_h - 7kv_l}{10} &amp;= \frac{3v_h-3v_l}{10}\\
7kv_h - 7kv_l &amp;= 3v_h-3v_l\\
k &amp;= \frac{3}{7}\end{split}$$

Thus, I suspect that your mixed strategy for the high type is given by the cdf $$F(s) = \frac{3}{7}\bigg(\frac{S-v_l}{v_h-S}\bigg)$$