Um, my teacup holds 14 ounces max. I was dumping way too much water to make it worth using. That was a few years ago, though, so smaller kettles might now be on the market. I have an electric stove, so I went back to using a stove-top type.
How much water are you trying to heat at one time? If it's just enough for a cup of tea, I know from personal experience that many kettles heat more than needed. I had a nice Hamilton Beach one, but the minimum amount of water you could heat in the kettle was about 1 1/4 quart.
Ages ago I had a coil-like immersion heater. You would put it into your cup of water, plug it in, and wait a couple minutes for the water to come to a boil. That might be an option.
Are you sure it is rust?
Teakettles tend to develop a mineral scale inside.
The scale can be reduced or removed by boiling a mixture made of equal parts water and white vinegar. Fill the teakettle with the diluted vinegar bring it to a boil and let it sit overnight. You will need to rinse the teakettle and may need to boil one or two pots of water to remove any lingering traces of the vinegar.
My grandmother always kept a glass marble in the teakettle to help reduce the mineral buildup, not sure if it did any good!
