If your daughter's having lessons, get her teacher to give your trumpet the once over.
It might be that there's an easy fix (like replacing the perished leaky cork on the main water key) but it's also quite possible that it's just a Bad Instrument.
It might also be that the instrument was put away for 20 years having not been cleaned first, in which case there might be an internal layer of grot which can significantly change the bore size. After all this time you probably wouldn't shift that just with a bendy brush and hot water, though you could have a go. Most music shops would send the trumpet off for a chemical clean, but you should get their advice to see whether it's worthwhile.
The differences between good and bad trumpets are often measurable as 1000ths of an inch (internal bore variations, tube wall thickness and so on) so even though two trumpets may look identical, the critical differences aren't visible to the naked eye.
And finally, it might be that your mouthpiece is no good - if there's no marking on it identifying its size then ditch it for a known quantity. A 7c mp is commonly the size provided with new instruments, but most players use a large mp (smaller number == bigger mp). You didn't say whether you tried your daughter's trumpet with her mp or her trumpet and your mp.
But realistically, $80 isn't very much to pay for a trumpet, which probably indicates the quality you should expect from that instrument.
Don't soldier on with a ropy instrument - if you can afford to upgrade then do - it's so demotivating when your instrument lets you down (valves stick, tuning inconsistent and so on). Of course, after you've paid a certain amount then these mechanical issues aren't so apparent, and paying extra buys you ever diminishing returns.