I didn't know this poem, and I've read hardly any Heine, so I'm pleased to read it. I think you're right that the simplicity of the language is a challenge here. That's not much leeway for alternative word choices. I had a couple of thoughts, though not on how to make it rhyme.
The first relates to 'kühl'. The day is hot and humid, as a result he's tired. The night is cool. Being both hot and tired, the night and its promise of sleep and its coolness (and lower humidity) seems to be something that's appealing, inviting -- a relief. 'kühl/cool' here can also mean distant, indifferent (he was cool towards me). But attractiveness/relief seems to be the primary sense of what's going on here, though both may be in play.
'frigid', on the other hand, is hard to see as appealing, inviting. It has sexual connotations, too, that go beyond indifference, connotations that I don't think 'cool' has and that I don't think the German has either (I'm very, very rusty, mind).
So why not stick with 'cool'? The main and secondary meanings are kept. "Death is the cool night" scans as a headless iamb followed by a double iamb and the next line is headless too. (I guess you can pronounce 'cool' as dipthong, but then it scans fine too: trochee, iamb, iamb).
"schwül" means muggy, hot and humid. 'sultry' as per your crib catches this but adds a sexual, seductive sense that I don't think is there in "schwül", and seems to suggest that the day is attractive (or trying to be). 'stifling' is a nice choice for the internal rhyme. It maybe adds a slight interpretation: suffocating, restricting, whereas the more neutral "schwül" allows the reader to make that association him or herself without leading them. 'Stifling' doesn't necessarily imply humidity, I think, though I might be wrong. 'close' might be an option somehow. It plays off the indifferent, emotionally distant night, I guess, though that would seem to over-steer the reading of the first line, it's meaning as 'near' is likely overbearing. I guess 'muggy' is a neutral option in the way that "schwül" is (again, I could be wrong), though it's a little underwhelming as a word (against which 'stifling' is maybe a little over-strong). 'humid' might another neutral option, but also seems a little weak and loses the heat. Sorry, I started this paragraph thinking I might reach a useful conclusion.
For the last line, I guess you could have
"The day has worn me out"
Which has a slightly stronger consonant rhyme with 'night' to my ear.
I think you need a comma after 'Life' to show the elision of "is".
Sie singt von lauter Liebe
can also be "she sings of pure love" (as opposed to "only" love -- she sings of love and nothing else). To make it scan, it could be something like: "she sings of perfect love" (or "purest love"). I don't know that this is better, but maybe it's an option.
Anyway, time for me to embrace the cool night.