How to increase the current in this circuit?
Your problem is base drive. Your controller GPIO is unable to provide adequate current, even though your are effectively shorting it to ground (I say "effectively" because 0.7 volts, the actual limit, is much less than 3.3).

If you look at a 2N2222 data sheet and look at the saturation curve at the bottom of page 494, you'll see that, for 200 to 300 mA, you can hope for a saturation voltage in the range of 0.2 volts ASSUMING A GAIN OF 10. In your circuit, with a 5 volt supply, 2 1.3 LED drops and 0.2 volts across the transistor, you can expect a current of $$i=\frac{V}{R} = \frac{5-2.6-0.2}{8} =  275\text{ mA}$$ This in turn requires a base drive of 27 mA, and your GPIO will never produce it. 

So, how to proceed? Well, you need to boost the current capacity of your GPIO pins, and you can do this with the following circuit







simulate this circuit &ndash; Schematic created using CircuitLab

Two extra transistors, rather than 1, are required to maintain the operating sense, in other words to turn the LEDs on with a logic high on the GPIO pin.

Of course, if you need an extra PNP anyways, there is no need for 3 transistors. You can do it with







simulate this circuit

This circuit will probably work well, but it's very slightly iffy in terms of total gain, and may not give quite as much current as you expected, but it will be much better than your first attempt.

The PNP can be any good signal transistor. I've shown both 2N2907 and 2N3906. Both will work fine.