Hi Gemma,
I would reccommend that you spend your time strengthening up the ankle and improving your balance before buying a brace. You don't want to become reliant on a brace for the rest of your life.
Have you been trying anything to reduce the swelling?
To be honest swollen and struggling to walk after 3 weeks I would consider a second opinion from the hospital. For most ligament strains of the ankle you are looking at 6-8 weeks for recovery back to activity. At 3 weeks I would expect no swelling and little pain unless twisting or turning.