In the big picture of water that needs treatment, water sources in Yosemite taste good regardless, so the filter is just for the very odd chance you would get giardia. Good idea to filter, but if the filter breaks, I would not hesitate to drink the water unfiltered, as long as you get it from a good flow away from the several outhouses along the trail.
There is a drinking fountain at the Vernal Falls bridge, but that is less than an hour up the trail. Sometimes it is out of order, so if counting on it, get confirmation from the ranger when you get your permit.
I personally prefer the Mist Trail to the JMT- it cuts off nearly a mile but you do have to go up/down a lot of steps and you get wet with the mist (which I think is wonderful- particularly nice coming back when it is hot. Go up one and down the other if you want to see the most. If you are going up pre-dawn, then the JMT is better with a head lamp. At least come down the Mist Trail.
And take the time to go to the overlook at the top of Nevada Falls and look straight down!
You should have some water sources above LYV, but it is looking like a dry year. The last 100% reliable water is the Merced River, just before the fork in the trail before you get to LYV camping area.