Because we don't have any other silver pieces, we also didn't have any silver polish in the house. Instead, I found a great and easy three step method online at Tater Tots and Jello.
1) Line the kitchen sink with tin foil
2) Add in about 1 cup of baking soda and fill with very hot water
3) Place pieces to be polished in the sink so they are touch the tin foil
You can literally watch the tarnish fade away. It actually came off our new tea pot in chunks.
It took us about 30 minutes to do all 5 pieces, and Jeremy had to change the tin foil and water a few times as we found it wasn't working as well after a bit. It was definitely much easier than trying to polish the thick layer of tarnish off though.