On most broms the adult plant usually doesn't die until it has made a baby or two. I was going to suggest checking carefully under and inside the lowest leaves, but you obviously already did. Well done!
Once you turn on your heating this fall, you might think of spritzing the whole plant with a little plain water or a very dilute pinch of fertilizer in a sprayer bottle. They love humidity and the leaves will stay looking nicer for a longer time if you "rain" on it a little bit. Pups will grow a tad faster with a tiny bit of fertilizer as well. I spray my broms maybe once a month with my orchid fertilizer (just because I'm out there spraying the orchids anyway).
