Ttam, over the next week or so keep an eye on your hookah. I don't know whether or not putting the stem in the dish washer will do any more damage than leaving a filmy residue and bad taste, but you want to watch for rusting, especially if it's a Chinese generic. Give it a good rinse with cold water as cold water cuts through soapy residue better than warm water.
And invest in a bottle of pure lemon juice or white distilled vinegar and a set of cleaning brushes if you don't have any at present. Hookah John sells a three-piece brush kit
for only $5.99.
Hopefully after a good rinse and a session with Nakhla Mint the taste will go back to normal for you.