Old faucet, new portable dishwasher?

you should be able to rebuild all of the faucet not sure where the the packing came out of.if it was from the neck it was from the pressure when dw is not filling or when it drains. if the packing came from the handle stem i would remove them and get replacements at your local hardware store. hope this helps. **********if you want to replace it that's good takes a good amount of work to replace back mount style faucet i Would take a look at the install before you make any decisions sounds like you have a pretty old home.or at least plumbing take a look and let know what you see I'll give you a better idea then. you can e mail me the site you are looking at i will have no way of knowing if the nuts on your old are the same threads as the new. send me that and if you can send me a pic of the one you have now i can make a better answer. the link did not work sorry