Hi Mike, I'm very sorry for your loss. I'm writing because I strongly disagree with some of the advice that has been given to you regarding how to proceed with your existing bird, and any bird you may bring in.
It is not wise to randomly treat with ivermectin and tetracycline in the absence of a confirmed diagnosis of illness. These drugs do have serious side effects, particularly the ivermectin, which is toxic to a bird's liver in too high a dose. Also, dosing one bird via the water method is not recommended when the bird can quite easily be dosed by hand. Dosing in the water is the least accurate way to dose a bird, and is generally used for large flocks of birds only. To dehydrate a bird by withholding water, and then offering medicated water is not a good method of treatment, especially when you would be treating for illnesses you don't even know the bird has.
Parakeets die suddenly for many, many reasons. Since no medical diagnosis was made on the death, let it go and move on. A lipoma, or fatty tumor, on the chest is a very likely cause of a young budgie death as lipomas are extremely common in budgies from inter-breeding. As the tumor grows, it presses on air sacs and causes restricted breathing. This places a burden on the heart, and eventually the bird dies. This is a much more realistic diagnosis than air sac mites, which actually, are not that common. If your remaining bird begins to exhibit breathing problems, take him to a vet (preferrably avian, or one that specializes in exotics) for an exam and diagnosis, and then treat as he recommends. Otherwise, you may well kill your existing bird trying to help it.
Yes, do get another bird for your budgie. I've had budgies for years and they are, by nature, very accepting of companions of either sex. You may try adopting a bird through a local bird club or bird adoption organization (these are usually health checked). If you buy another bird, pick an active, healthy one and yes, keep it separate for a while for observation before introducing it to your remaining bird.
The best diet for budgies is a quality small pellet such as Harrisons, a quality seed mix, such as from Golden Feast, and a daily mix of fresh spinach and romaine lettuce, broccoli, a few kernals of plain frozen corn, a little grated carrot. Budgies don't usually take to fruit. Do not feed iceberg lettuce, or lettuce other than romaine, as they cause loose droppings and have little nutrition. Always keep a cuttlebone in the cage, or a mineral block with iodine. Budgies require iodine (not the liquid kind--that's poison) such as the kind in iodized salt, or they will develop thyroid problems. If you feed a quality pellet in small amounts, that will have iodine in it.
Your birds will need flight to remain healthy. Try to put them in a cage that is at least 3 feet side to side, 1 and a half feet to 2 feet deep, and tall, so they have to fly up to their perches (good exercise!). Put their food on the bottom of the cage on some paper towels (not beneath perches, or droppings will contaminate it) so they have to fly up and down. You don't need an expensive cage for budgies, and there are ones like I've described, with thin bars that have 1/4-1/2 inch spacing that are not at all expensive. Also, give your birds a variety of small toys. It's best not to breed, as a variety of problems arise from breeding budgies, including eggbinding and splayleg from improper nesting--be happy with finding a friend for your bird.
Most budgies hand tame easily, with patience. They often do not like to be petted (fingers are big, and they're delicate), but they do enjoy sitting on your finger, or riding on your head or shoulder. Hand tame them by NOT trying to touch them. Offer you hand with a romaine leaf pressed between your fingers, or some corn in the palm of your hand. Have patience. Let the birds come to you, nibble, and go away, without interference. Once they trust your hand not to do anything scary, everything moves quickly.
Finally, keep up with nail clipping (clip the sharp tips only, and be careful to avoid the vein visible in the nail or the bird will be hurt, and bleed--a crumb of constarch on the tip will stop bleeding), as they can get caught up in toys, etc. if the nails are long and curl under.
Good luck! I hope this helps. You obviously love your bird, and I wish you the best of luck in finding a healthy companion.