Gym bikes seats are similar to beach cruiser seats, not made for cranking along at 15-20mph. I'll find something else to do besides going to a gym on a rainy day; I can't stand the indoor environment, especially the loud music.
1) Find a hill and do some repeats up-and-down it - either a paved street or even a dirt rise, and run it or briskly walk up and down it.
2) Find a parking garage and use the stairs - 'repeats' up and down a stairwell, skipping a step as you go up, or go from one stairwell to another. Stairs are kind of boring, but I'd rather do that than use a gym.
3) Just go out for a jog in the rain. Always fun. IF you're not cold it won't make you sick.

But if you do have to go to a gym try using the stairmaster of equivalent device instead of a bicycle. Great cross-training.