On are not syllables
The 5 / 7 / 5 count is in on, which are sound units roughly equal to a short vowel with or without a following consonant. Japanese ん counts as one on even though it is not a syllable you can stress. English has no equivalent unit, so any English 5 / 7 / 5 written in syllables is already an approximation — which is fine, as long as you know that is what you are doing.
The specimen above counts 5 / 7 / 5 in syllables: cold-kit-chen-win-dow, the-ket-tle-clicks-it-self-off, be-fore-I-stand-up. If you counted it as on, the second line would be eight. The honest position is that English haiku are written in a five-seven-five shape, not in on.
The kigo does more work than the image
A kigo is a season word — a word whose presence places the poem in a season without any further explanation. Frost, swallows, watermelon, first heating oil. It is a compression trick: three syllables of season, and the reader supplies the temperature, the light, the length of the day.
In the specimen, cold and kettle between them place the poem somewhere in late autumn or winter without naming a month. A haiku without a season word is not broken, but it loses the free information that the kigo carries. When you draft, put the season word in first and build around it.
The cut, and what goes in the gap
A kireji is a cutting word — in Japanese, a particle such as ya or kana that suspends the sentence. English does not have them, so translators use a dash, a colon, an exclamation, or simply a line break. The cut does not connect the two parts; it holds them apart so the reader has to do the joining.
Here the cut after window sets a cold interior scene and then hands over a small domestic event. Because the two are not joined, the reader supplies the cause: nobody wanted to get up. That inference is the poem. Remove the cut — write the kitchen was cold so I stayed in bed — and there is nothing left for the reader to do.
Two lines that fail the form every time
First, the simile ending: my breath like smoke. It closes the image by explaining it. Second, the abstract last line: and I feel alone. Both tell the reader what to think. A working ending usually lands on a physical object that was already in the room.