Training for a backpacking trip

Hello!

I am looking for advice and suggestions on training for a backpacking trip. In short, how would you prepare for short but rather intense a 22-mile-long 2-3 day backpacking trip with 10,000 ft elevation gain/loss?

Some background on my situation: This trip was just decided upon and is coming up in 12 weeks. I hike fairly often and have done a few backpacking trips spanning 3-5 days in the not-so-distant past. I backpacked through the Grand Canyon for five days this year (gaining 5,000 ft on the climb out alone), but I am admittedly totally clueless when it comes to fitness outside of events like this. My upcoming trip involves a lot more elevation change in a shorter distance, and I feel that, while I could probably physically get through it now, I need some pretty serious strength training and cardio to fully enjoy the experience. I currently have access to a staff gym that has some free weights and an elliptical. Hiking regularly would be ideal, but I live in Vermont, so my days doing that are limited this time of year. I am open to joining a gym with more equipment if that would make a significant difference in my training.

Again, any suggestions would be greatly appreciated, as I've never really had a fitness routine to speak of and am completely lost when it comes to strength training or training routines in general. What would your training program look like if you were preparing for this hike- specific exercises, number of days per week, etc?

Thank you for your ideas!