Hello, this is my first plumbing project and I'm not sure what everything is called, but I could really use some help figuring out the parts I need.

I've just replaced my kitchen faucet, but the fitting that connected the hot water from the faucet to the valve under the sink had a cracked/broken gasket.

The broken fitting - front:
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The broken fitting - back:
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The spiral metal tubing that goes from the valve and was inserted through the back of the fitting:
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What is this kind of tubing called? I tried googling just about every variation of adjectives I could think of and couldn't find it...

I went to several stores, and none of them had that exact kind of fitting. Does anybody know what it is, or what other kind of fitting I could use with that kind of tubing?

At first I thought the fitting was one piece, but I was able to take it apart into a compression nut, a washer, and the (broken) gasket. I found a fitting that looked similar at the store, but it's not quite the same and the tubing didn't fit through the back of it.

Original fitting (top) and new fitting that's similar but doesn't work (bottom):
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If I can't find the right fitting, I suppose I will have to replace the valve. Of course, since this is my first plumbing project, I'm not sure what kind of socket the valve is. The current one looks like a 1/2" pipe thread?
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Thanks for any help!