Mm I want to increase my kicking and punching speed for my tae kwon do. Other websites are either not clear enough, or it involve equipments or being in outdoors. Is there any traning that I can do to to increase my kicking and punching speed in my room??
Or equipments that I can use like ankle weight (I heard its not good for your back.. )?

Do you have a bad back? Are you healthy enough for a ruggid work out.

First practice, practice and practice, pushing for more and more speed.
But beyond that technique has to be practiced getting it down and all of the moves smooth. Speed along does little.

That depends on your goal. If you are performing Kata and want to be fast and smooth then just practice the Kata over and over and over again.
If you are sparring and want to be hitting harder and faster then you will NEED equipment. You need 16oz boxing gloves, a heavy bag, and a good set of shin and instep pads. Then you work the heavy bag fast and hard for 20-30 minutes every day.
The best way to get better at punching and kicking is to punch and kick.

And for reference I'm a 1st Dan BB in taekwondo and I'm currently taking Brazillian Jiu Jitsu classes at an MMA gym.