Rhinoplasty for the broken nose
A rhinoplasty procedure can accomplish straightening the broken nose and shaving down the dorsal hump that has formed across the bridge line. All of this can be accomplished with a closed rhinoplasty approach with all the incisions placed on the inside of the nose. For more information and many examples, please see the link and the video below
With your history the problems you describe are probably secondary to your nasal trauma. You need an examination to determine what can be done to improve both shape and function.
It is possible your nose was broken and both the cartilage and the bone affected. Injury to the cartilage inside your nose can lead to a breathing blockage. In order to modify or change the appearance and improve function you would have to consider a Septorhinoplasty. This should only be done by someone very experienced in this particular operation.
I suggest seeing a Rhinoplasty Expert who can help guide you. You could have a great result with a scarless closed rhinoplasty to shave down the profile, sculpt the tip and narrow the bones. Depending on the internal septum, whether or not its deviated, would indicate whether you need a septoplasty in addition. I recommend finding a surgeon who dedicates their career to rhinoplasty. All the best!
Broken nose, what happened?
When people talk about a broken nose they almost always think of the nasal bones being fractured. This is very common, but there's another way a nose can be broken. The lower 2/3 of the nose is made of cartilage and it too can be broken. This is a common cause for a deviated septum. Here it appears from the pictures your tip cartilages have been twisted. One point of the tip is lower than the other and the tip turns to the right. It's even possible to consider the width of your tip a result from the injury, although this may well be just how the nose grew. You can see (and certainly feel!) the edges of these cartilages because of the thinness of the skin. Finally the very end of the septum may or may not be contributing to the twist. This is easy to tell by examining the nose.