Why are liquid ring compressors not used in a jet engine?
You said to ignore the cooling of the liquid, but that's something that cannot be ignored.  With the compression ratios achieved in typical jet engine compressors, the air leaving the last stage of the compressor may be 800 - 1000 deg F.  You won't find a lot of liquids that won't just completely vaporize at those temperatures.  Yes you could potentially cool the liquid, but the cooling apparatus would add a huge amount of weight, as well as consume a lot of power.