"My son, peace be unto thy soul; thine adversity and thine afflictions shall be but a small moment; And then, if thou endure it well, God shall exalt thee on high; thou shalt triumph over all thy foes." (D&C 121: 7-8) 
 
There are many reasons why people face trial and affliction in this mortal existence, sometimes our afflictions are the consequence of poor decisions (such as being injured in a car accident while driving drunk) and other times it is the result of some thing completely out of our control such as genetics or war.  
 
In either case these afflictions and trials are part of our progression, they give us opportunities to learn and grow and as the Lord told the Prophet Joseph Smith while unjustly imprisoned without due process or trial in Liberty Jail, our afflictions are for but a short period of time, if we endure well we shall conquer our tests in this life and obtain eternal life in the Celestial Kingdom with our Heavenly Father.  
 
God is omnipotent he knows all things and this is why he allows us to face challenges because he knows that without trials and challenges we cannot grow and become more like him.  
 
In our Church it is taught that every trial is an opportunity to grow and if we place our faith in God he will guide us through even the darkest of hours.  
 
Also he will heal every amputee in due time, for many this will not happen in this lifetime but in the resurrection when spirit is reunited with a perfected body of flesh and bone just like the body of our Heavenly Father and Jesus Christ.  
 
In all things faith is the key, know that our afflictions are but a moment in the eternities and our reward for enduring them well far outweighs the price paid.  
 
A good analogy is working out, while working out we become tired and sore, we suffer discomfort at times while working out. This discomfort though is the muscle tissue being broken down so it can be rebuilt better and stronger. In the end he increased health and strength from working out far outweighs the temporary discomfort of working out.  
 
The same goes here our afflitions and trials are discomforting but they are there to tear us down a bit so God can rebuild us stronger and better than we currently are through faith in him. In the end the discomfort of our trials are outweighed by our progression and additional strength gained from facing them.  
 
Thank you for the question I hope my answer helped!