Lizz;
Strength training is a good answer to increasing muscular strength quicker, than just with specific training (cycling). I would suggest 2 options:
Weight training, I would look at a routine where you lift 70% of your max weight, and keep in the 8-12 rep range. Doing 15 or more reps with weight training isn't as effective as regards building strength, as opposed to lower reps and higher weights. As you are using this as additional training for your specific training, you want to work towards muscular strength (up hills, and sprints), the endurance aspect is covered by your cycling. Exercises such as squats, hack squats, leg extensions, stiff-legged dead lifts, leg curls, and calf raise; are all good. Squats are probably the most effective, but it takes some learning to coordinate the movement. Remember to balance your workout between your quads and your hamstrings - if you are cycling and strength training you run the risk of over training your legs, or overdeveloping one area. Quads tend to develop more readily than hamstrings (even with just cycling), so you want to "balance" this development in your hamstrings, if you do not you can see injuries such as muscle tears (hams).
Rowing machine. You mentioned you have little time to add additional training, so I would suggest getting a home rowing machine. Rowing with the machine set at a high tension will develop your hams and quads (as well as your upper body). I would suggest something like 4 x 5 minute "sprints" (rest in between), this will act as a strength building exercise. It will also help you balance your ham/quad development and reduce the risk of unbalanced development. If you experience a general muscle soreness that lasts for more than a few days, you are over training, and you need to add more off training days.
Hope it helps,
-CG