I finally finished my first transistor circuit and looks as follows:
For the LED to light up properly, I assumed a forward current of 70mA and a voltage drop of 1.3V. The datasheet of the PN2222A states that at 70mA collector current, the saturation voltage \$V_{CE}\$ will be about 0.06V. So $$ R2 = \frac{5V - 1.3V - 0.06V}{0.07A} = 52\Omega $$ Lowest value of \$\beta\$ equals 10, resulting in a total minimum current gain of $$ \beta_{Total} = \beta^2 + 2\beta = 120 $$ \$I_{B}\$ is therefore $$ I_{B} = \frac{I_{C}}{\beta} = 583\mu A $$ So $$ R1 = \frac{3.3V - 1.62V}{583\mu A} = 2882\Omega $$ When I plugged in both voltage sources, the LED only glimmed lightly. I measured \$I_{C} = 42.4mA\$ which is definitely too low. For troubleshooting purposes I checked every voltage drop and one was far away from its theoretical value: \$V_{CE} = 0.7V\$.
Why is the real saturation voltage way higher than stated in the datasheet?