If it continues to blow cold air you may assume the problem is solved. Kwik Fit are not a/c competent. It sounds like the last firm know their stuff. They have invited your return if it blows warm again. Which is what a competent firm would say.
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My 10yr old car ac blew warm air then kwit fix, re-gased it and it started blowing warm air in about a week then took it back to re-gas and the same thing happened. Kwit -fix said it couldn't repair leak and never put dye in but they didn't detect any leak in the ac system. I asked for a refund and went somewhere else. There claimed to be car ac specialist. They sucked out old gas and put dye in using hi-tech lightin checked compressor, hoses, seals and everything, then re-gased it and the pressure held on. Car remained with them for 3hrs. They rang me to pick my car I was worried that compressor had blown but they said they dected no leak but I should return it if it blew warm air. Nearly two wks now it is stil blowing icy cold. .which far exceed the first repair time. My qustions are: What went right? Does it mean everything is ok? Will it blown warm or is it fixed now? Any expect view would help. Thank u my family.