How can I calculate the base current of a transistor to get a specific emitter-collector voltage?
There exists a method for determining the saturation voltage \$v_{CESAT}\$ for a given current \$i_B\$.  With a few assumptions for the forward and reverse current gain \$\beta_F\$ and \$\beta_R\$ , we can determine the current \$i_B\$ from the saturation voltage \$v_{CE}\$.

I've referenced chapter 5 page 233 in Microelectronic Circuit Design by Jaeger and Blalock ISBN 978-0-07-319163-8

Assume that both junctions are forward biased. (This is required to approximate \$i_B\$ and \$i_C\$ in order to arrive at this equation).

\$v_{CESAT} = V_Tln(
\frac{1}{\alpha_R}
\frac{1+\frac{i_C}{(\beta_R+1)i_B}}
     {1-\frac{i_C}{\beta_Fi_B}})     \$ for \$i_B &gt; \frac{i_C}{\beta_F}\$
This equation is only valid in the condition condition that \$i_B &gt; \frac{i_C}{\beta_F}\$ 



  This equation is important and highly useful in the design of saturated digital switching circuits.  For a given value of collector current, [this] can be used to determine the base current required to achieve a desired value of \$v_{CESAT}\$.


With \$V_{CE} &lt; 0.7V\$, it can be assumed operation is in the forward saturation region. For your situation with \$V_{CE} = 220.745mV\$, we can simplify to 
\$6835 = \frac{1}{\alpha_R}
\frac{1+\frac{i_C}{(\beta_R+1)i_B}}
     {1-\frac{i_C}{\beta_Fi_B}}\$

But, we still need to know \$\beta_R, \beta_F,\$ and \$\alpha_R = \frac{1}{\beta_R+1}\$

Choosing arbitrarily, and within reason, \$\beta_F = 50\$ and \$\beta_R = 1\$ we can solve for
 \$i_B = 0.39414mA\$ with \$i_B &gt; \frac{i_C}{\beta_F} = 0.39116mA\$
And with a similarly reasonable \$\beta_F = 100\$ we find 
\$i_B = 0.19850mA\$ with \$i_B &gt; \frac{i_C}{\beta_F} = 0.19558mA\$

That is great and all, but how do we know these results make sense? If you are computing \$V_{CESAT}\$, you expect a small value close to zero, even \$0.06V\$ is sensible. Plugging the currents back into the original equation in both cases it yeilds \$0.220745V\$  

Some methods to check the work cannot be used without a value for \$I_S\$.  Also note that the above equation is independent of \$I_S\$. 
Final note, the above work was done assuming npn transistor, but pnp can be computed as well.