I'm pretty sure it's a weak joke calling back to a previous reference in the speech.
But why, some say, the moon? Why choose this as our goal? And they may well ask why climb the highest mountain? Why, 35 years ago, fly the Atlantic? Why does Rice play Texas?
We choose to go to the moon. We choose to go to the moon in this decade and do the other things, not because they are easy, but because they are hard
"The other things" here refers to climbing the highest mountain, flying the Atlantic, and Rice University playing [American college] football against the University of Texas. Rice and Texas have a long-standing football rivalry (now over a century old); "Rice playing Texas" is a challenge of particular resonance with JFK's audience at this speech. That brings us to the passage you cite:
because that goal will serve to organize and measure the best of our energies and skills, because that challenge is one that we are willing to accept, one we are unwilling to postpone, and one which we intend to win, and the others, too.
Assuming the same set of "others" here, this is superficially a puzzling construction. The challenges of climbing the highest mountain and flying the Atlantic had already been won, so it's nonsensical to talk about "intending to win" them. That leaves only Rice playing Texas, so my interpretation is that JFK is jokingly saying "we [the United States] intend to go to the moon, and we [his Rice University audience] intend to beat Texas."
Here's the printed script for the speech:
"Why does Rice play Texas?" has been handwritten in (possibly by Kennedy himself) -- a late addition to the text -- and the phrase you're asking about isn't present at all, strongly suggesting it was ad-libbed as a callback to the Rice/Texas reference, which might explain why it's so grammatically clunky.
At the time of the speech, Rice and UT were 5-5 over the previous decade despite UT being a significantly larger school. JFK's speech doesn't seem to have brought Rice much luck; Rice and Texas' next game, a little over a month after the speech, was a 14-14 tie, and in the forty-odd games between the two since then, Rice has only won twice.