You could try bringing it to an upcoming repair cafe:
but that sounds like a tough problem to solve. It's very likely the whole pot has corroded/thinned but it just appears to have a few small pinholes in it. To really solve this you'd likely need to remove all of the oxide/calcium from the inside (which will likely eat more of the metal while it's at it), then weld more copper into the pin holes until it was water tight (which will cause warping from the heat distortion), then re-plate the inside of the pot to rebuild thickness - might be time to retire the teapot to a nice spot on a shelf :-)
Chris