Hey guys. I've been doing karate for 2.5 months now. I took my first belt too

... The problem is that my group mainly consists of young children and the training is not so hard. By that I mean that I have not developed much leg power or improved my body that much (of course I've improved my technique and flexibility). My friends back where i used to live, take karate pracice too but their group consists of people 15 years old+ so the training is harder. Any tips for improving leg
strength and
speed on my own? I want to start doing skipping rope too but i don't know what to use as a rope. I tried using my karate belt but it's way too short. Any opinions? Also are there any exercises I could do AT HOME for leg improvement?(I would like to improve my flexibility too)