How to connect a 1mA IR receiver with a 20mA led in the same circuit?
Typical IR receiver needs a supply voltage between 2.5V ... 5.5V. Therefore you cannot use the 9V battery directly.

As for the circuit, you could try one of these circuits.



You can use any small-signal PNP you have in a TO-92 package.  

And without any resistor, my IR receiver gives 15mA max of current through the LED at 5V supply. This is why I skip the resistor in the first circuit. But for safety maybe you should add one.   

EDIT 

I played with this circuit and I add a capacitor to make a diode to light longer.

  

Because without this capacitor diode lights-up for a very short period of time.