Explain this sentence structure

Hi all,

Can someone explain this sentence structure for me?

水は冷たいことは冷たくても、氷のように冷たくはなかった。そしていやな味がした。
The water was cool but not icy, and it had an unpleasant taste.
mizu wa tsumetai koto wa tsumetakute mo, koori no you ni tsumetaku wa nakatta. soshite iya na aji ga shita.

I am quite confused the wording, or rather, the structure of this sentence. It's very rare that I see two はright next to each other like that in a sentence.
[水は冷たいことは冷たくても] - The water is cold, but
[氷のように冷たくはなかった] - It was not cold like ice.

What if I wrote 水は冷たいけど、氷のように冷たくなくて、味がまずい（まずい味がする）。
Would it still make sense? With my level of Japanese right now, this is exactly what I would write or say instead of the original sentence. The [冷たいことは冷たくても] part feels redundant and confusing to me. I understood it as "the water is cold thing but not cold".
And [冷たくはなかった], why is there a は in there? What purpose does it serve?
Is there another word to describe cool, as in cool to touch?

Thanks in advance!

The tone is something like:
- The water was cold alright, but it was not as cold as ice. And it had an unpleasant taste.
- (If you ask me) I would say the water was cold, yes, but not as cold as…
This “…koto wa … temo/kedo” is an expression to express contrasting situations/showing a lower degree of something/hesitation (in intentions). Let me try to come up with some examples later…

The sentence you wrote is OK, yes, it conveys the idea, except that the original sentence is in the “ta-form.” Do you want to give it another try?

The “wa” is in there to highlight that it was not “as cold as ice.”