What exactly is the lack of treatment that you have escalated to the hospital on 6 October? It's not really clear from your post.
High blood sugars and dka can cause an adverse impact on mental awareness/cognition.
Damage to the nerves around the wind pipe, so the vocal cords and gag reflex, is a well recognised risk of tracheostomy, especially emergency tracheostomy, it doesn't necessarily mean the tracheostomy he had was done wrong.
What treatment do you think should have been given to the amputation site, other than applying dressings, bearing in mind he's not fit for vascular treatment while the tracheostomy is in situ?
Sometimes it is necessary to amputate to manage gangrene, sometimes even after initial amputation and all correct management it can be necessary to amputate further, especially with an established sepsis. The further amputation can just, unfortunately, be necessary, and not anything to do with a fault in care.
Severe anemia is treated with blood transfusion.
Have you had any discussions with his managing consultant so they can explain the treatment course to you, and your brother?
Wishing your brother a speedy recovery.