Powering an LED with transistor in saturation mode
In order to get the base current IB, I looked for DC current gain in
  the datasheet. The lowest value is β=10 as shown in Figure 11.


Figure 11 uses Ic/Ib = 10 to guarantee the lowest possible saturation voltage, but the transistor will still have a low saturation voltage at higher ratios. In figure 4 you can see that a low saturation voltage is typically maintained for Ic/Ib ratios as high as 50 at Ic=150mA. 



The PN2222A is guaranteed to have an Hfe of at least 50 at Ic=150mA and Vce=1V, and typical Hfe is ~200. To switch 70mA with 2mA you need at least 35. So even in the worst case it should be able to do the job, just with  a higher saturation voltage than you used in your calculations.