A reasonable way would be to debug the hard- and software by checking it's operation step by step. If the original software doesn't provide test functions (I don't see any a first sight), then it should be modified respectively. You can start with simple things:
1. Make the LED blink unconditionally
2. Make the LED blink at key press
3. Output some test text to the character display
4. and so on
Obviously, suitable test steps depend on which test tools are available to you, e.g. a multimeter, an oscilloscope ...?
At least the assembly text style looks good, so as a first guess, I would rather suspect the hardware setup.
By the way, can you be sure to have transferred the Hex image correctly to chip?
Finally: You decided to do a 8051 assembler project. Then you should learn 8051 assembly language.