how to implement an integrated rechargable battery in my project?
Your best bet is to use a battery management IC.  Searching Digikey for "Battery management" and then filtering by battery chemistry and then price, gave me the TSM101/A.  For less than a dollar, it seems to be capable of what you want.

Just replace the mains and rectifier portion of the circuit with a wall wart.  There are some examples with charge status LEDs.

Here is the data sheet.