There are many I2C adaptors and they all look similar. Goto Ebay and see if you can find a photo of your ACTUAL adapter. Then post a link to the Ebay photo.
Although the common lcd method()s seem to be the same for most "liquidcrystal.h" libraries, the constructors and backlight() methods often have different syntax.
Once you have shown the photo of your adapter, someone can point you to the universal "LiquidCrystal.h" library and the correct constructor.
Why do you have two spurious flying leads going to the "LED" pins where the enable jumper is supposed to be?
The library you cite appears to be for a quite different "backpack" and therefore not recommended:
It is then no surprise that it is not working if the only descriptor you are using is:
LiquidCrystal_I2C lcd(0x27,16,2); // set the LCD address to 0x27 for a 16 chars and 2 line display
We can only recommend that you use the more familiar fmalpartida library (and remove any and all other LiquidCrystal libraries from your IDE path) and then use bperrybap's "i2cLCDguesser" script to determine the correct descriptor to use with this display and backpack, which descriptor you then copy into whatever sketches you want to use.
But my question is - why do you have those wires connected at all?
The backlight LED is switched on and off by software command (when you master the code) - there is no reason to do anything other than leave the jumper in place over those two pins. It is just a novelty that the ability to disconnect the LED by a jumper is provided on the board.