STI Shifter Lever body loose
There are a few reasons the brake and shift levers could have become loose with respect to the shifter body, but none of them are repairable. Shimano shift units are essentially riveted together and spare parts are not available (to my knowledge).

This is a Old Ultegra ST-5500 unit I had, similar to a ST-2300  



The levers are fixed to the shifter body by the lever pivot pin. This may  have broken, or come loose from the shifter body. The lever pivot return spring may have broken, which would allow the levers to wobble on the pin. 

The levers also pivot laterally on a shaft in the shifter mechanism, this may have broken, or the return spring has failed.

The fix is to find a replacement unit. You should be able to find a left or right hand replacement unit in Ebay, Amazon or other online source,