While I'd obsessively use my ingenuiTEA over anything else you seem to have a lack of space to clean something like that. I'd recommend using the tea filters ($3 for 100) and make your own teabags. I used to tie mine shut until I ordered from upton tea imports to get a thermometer and a tea filter clip which snaps the filter shut and has a chain to pull it out of the cup with. Put water in cup add filter, steep, remove (but never squeeze the bag, it will release bitter flavors) and throw the filter away
Life is like a cup of tea, savor it slowly or it will be gone too fast