Designing an LED Driver
You have made a small mistake when computing the \$I_2\$. You say \$I_2 = V_2/R_2\$, which is not correct.

You should say \$I_2 = (V_2-V_b)/R_2\$, since \$V_b\$ is not equal to 0.

Then, you can say \$V_b = V_{be}\$ and assuming you will have the transistor in saturation \$V_{be}\$  is actually known and is equal typically to 0.7 V.

Moreover, there is also a mistake in the collector side calculations. More particulary you have thought the voltage drop across \$R_1\$ wrong.

From Kirchoff's law you can write \$V_1 = V_{D1}+I_1 \cdot R_1 + V_{ce}\$. In this equation you are missing \$V_{ce}\$. If you assume the transistor will be in saturation it is typical to assume \$V_{{ce}_{sat}} = 0.2V\$ (this value is given at the datasheet of the transistor). Now you can calculate \$R_1\$! Note this is the complete formula. The way you did it is also correct, since \$V_{{ce}_{sat}}\$ is quite small. You just make some approximation by ignoring it.