You didn't need mods for this! Just follow the steps
1. Open your Minecraft world
2. Give yourself a command block by /give @p command_block in the chat. You need to allow your cheats, if they are not allowed, just click "esc" >>> Open to LAN >>> Allow Cheats:On >>> Start Lan World and you are ready!
3. Paste the command from the link above into the command block and power it with button or redstone block/torch
4. Enjoy!

OK, you are done with installing the command. Now let's make fun!
To make the Broom, just throw at one place these items:
1x Stick
1x Blaze Rod
1x Diamond
1x Hay Bale
And you are ready! You will get an egg.
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To summon the Broom, just simply right-click with the egg on the ground and it will be spawned. To ride it, just right-click with a hand (without eggs or something) and you will ride it. To move it, use your W A S D buttons (to see how to make it fly, just continue to read)
The Broom will rotate in the directions North, West, East and South, but when you look between these directions (When you look at north-west, south-east..) it will be like in the second picture, so why I call this Broom "Gummy Broom" :D (look at the second picture below)



Gummy Broom *YAY*



When you are riding it, in the chat it will display a couple of commands:



To use them, just open your chat and click on the command, you want. You can turn on/off sounds, particles and the fly mode.
Fly Mode On - You can fly, you control your broom with your mouse (when you look up, you will fly to up, when you look down - you will go to the ground and etc.)
Fly Mode Off - When you summon the Broom, the fly mode will be off. With fly mode off, you control the Broom with your W A S D buttons (just the controls are the same with the horse's), but you CAN'T fly, so..
To turn off the fly mode without moving your Broom, just look straight down and click the button in the chat for it.
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And finally to destroy the Broom, drop a blaze rod on it. The blaze rod will NOT be deleted, so it didn't cost too much, but it will not give you an egg, only leather and saddle (a perfect saddle farm, huh?)

DON'T SPAWN MORE THAN 1 BROOM, IT WILL HAVE SERIOUS BUGS!!