It is evident the converter works, since you are getting a higher output than the battery V.
The diagram shows a coil rating of 200 mA. This may be a ballpark figure for the amount of current to expect. The schematic shows the output is rated 10 mA at 5V. To do this the 1.2 V battery must provide about 100 mA peak each cycle (at 20 kHz, demonstrated in a different boost converter simulation).
Although you are not drawing the entire 10 mA shown in the schematic, are you certain there is sufficient current going through the coil? A multimeter may or may not be able to measure this directly.
Can your battery provide 100 mA, perhaps as much as 200 mA?
Also if the operating frequency is too fast, it does not allow time for sufficient current to build in the coil. (However the hysteresis action is supposed to automatically adjust the frequency to compensate.)
Also a plain silicon diode may not perform quickly enough to conduct sufficient current during a cycle. (Again the hysteresis action should make up for this by reducing the frequency.)
The specs for 2N2222 look as though it should work fine.