Use File>place instead of pasting.
Unfortunately that's what I've been doing.
...all of the kerning is removed.
Not really sure exactly what you mean by that. Are you referring to manual kerning you did at some point, or are typical kerned pairs displaying some kind of odd behavior? Can you post a screenshot?
Hmmm, well I've reproduced it.
Can't explain it.
But I reproduced it. It's not a matter of "kerning" but rather the exclusion of the regular space character.
I tried converting table to text (in Word), then converting it back to a table before placing, and the spaces were preserved, but aside from all the space characters having a "custom tracking" value applied, the round-trip conversion resulted in a mangled layout anyway, so it was really only good for experimental purposes in the end.
In my standard workflows, I rarely if ever Place text-based content, and it's likely because scenarios like this were involved in the evolution of my habits. Word tables and InDesign tables have very little in common, and I would probably bring in plain text and reconstruct the table(s) in InDesign, if this was my project, without having even tried to Place. Actually, I've had better luck (copy/paste) with Excel to InDesign tables.