If he has been favoring the leg for over a week it is unlikely to be a simple sprain. I assume during this time you have been restricting activity to allow healing. Dachshunds are prone to intervertebral disc disease (rupturing a disc) leading to lameness, weakness, sometime pain, and neurological abnormalities. This breed is also prone to problems with luxating patellas--this is when the knee cap can move out of place. There are other injuries that can affect your dog. Your dog is in pain or he wouldn't limp or hold the foot up when he runs. In the very least he needs pain medication. Do NOT give any medications at home--dog doses are different than human doses and not all medications are safe for dogs. Also depending upon the problem certain types of medication are not appropriate. If we do not know the cause of injury it cannot be treated appropriately.