This works on glass coffee pots to get out the stubborn stains. Might be worth a try for your issue.
- a couple of cups of ice
- half cup or so of vinegar or lemon juice
- couple tablespoons of table salt
Exact ratios are not super-critical, as far as i can tell.
Dump this stuff in your vessel, and make swirly-swishy motions so the stuff sloshes over the dirty areas. Should come clean in a short time.
Rinse with clean water. Sip a homebrew.