Set aside some time to be alone at a keyboard or piano. Turn off the TV, radio and any appliances that make noises; the environment should be as silent as possible.
Open the sheet music you are familiar with and look over the notes. Try to memorize the note progression, imagine hearing the notes and internalize the tune. Try not to play any notes on the piano until you have read though all the notes in the song and have an idea of how the whole song sounds. Each stanza and line needs to be internalized before moving onto the next stanza and line.
Once you have internalized the tune of the song, the tone of individual notes and how all notes combine to play the music, sing along. Do not stop if you make a mistake; keep going straight through the piece. This step helps to check how well you did with the previous step of figuring out how the song sounds, stanza by stanza.
After you have associated a tone to each note, move onto the unfamiliar song. If you are unable to differentiate the sound of each note, do not move onto the unfamiliar song until you can hear in your mind the difference between C and D or A and B.
Look for notes that match in both pieces and use that as a starting point for learning the new song.
For the first couple of times through the song, do not play the piano but simply look at the notes and try to remember what that note's tone is.
Once you are pretty sure you know the way the music progresses, go to the piano and play and sing straight through the song, without any stops.