I would put a mirror into the cage if there isn't one
and see how that goes.  If there is one already.
I don't think it is a bad idea to get another keet.
One of my keets actually chewed off her
own toe out of loneliness.  She sqwuaked like hell when
the new one was placed in the cage but she never chewed
her foot.  Yes, things were a bit hostile for a few days.
Please make sure that there are 2 seed cups and 2
water cups in the cage as birds are territorial.  Also,
I have found that new keets do so much better when you
don't box them.  While the weather is nice take the covered
cage to the pet store and have the new bird put right in.
HOWEVER, before anything else, clean the cage
a day or 2 before you get the new bird so the transition
will be easier (no old scents) and there will be no additional
stress while the the birds are getting used to each other.
the adaptation goes on.  GOOD LUCK!

I am so sorry about your loss.  If you decide to get another bird, please keep it in a separate room for 30 days to make sure that it has no diseases.  I was always buying a new friend when I had a bird left alone, until my former love bird.  He was about 8 when my cocketiel died.  I decided to see how it went.  He was fine and lived until the age of 14.  His cage was not far from a window and he spent lots of time talking with the birds outside that would come and visit. 

Thanks for all your help. You're all of real help. So far the living keet seems to miss his mate. He seems depressed, he's nearly always looking at either the bare wall or the mirror. I tried to comfort him but he still doesn't fully trust me. If I slowly near my hand towards him, he gently nibbles my finger, pauses, and repeats the nibbling. If I try to touch him, he jumps back. This may take a while.........

Parakeets are very social animals who live in huge flocks in the wild so it is always better for them to have, at least, one companion so by all means, do get another one.   Having said that, you need to make sure that the surviving parakeet and the new one are not sick so you do need to quarantine the new bird in a separate cage for 30 days. During these 30 days, you should put them both on Ornacyn Plus (you can get in PetSmart, Petco or any big pet store).  Remove their water cup or bottle in the evening and give it back to them with fresh water in the am with the medicine in it, they'll be good and thirsty and take a good size gulp of it.   Give them fresh water with fresh medicine every morning for a week, rest for two days and then do another week.   You should also medicate them for parasites (in case the hen did have air sac mites which are contagious).  Use Scatt or, if you can find it, Ivermectin.  They are both water soluble, also and you just need to follow the instructions on the bottles as to the dosage.   Give it to them the first day, then do the antibiotic for a week, again, then another week of antibiotic and one more time.  You need to do it three weeks in a row because that's the life cycle of the mites, the medicine only kills the adults and the larvae and not the eggs so once the eggs hatch, you need to do it again or they end up same as before.  Do not, under any circumstances, take your bird to the store and have them put a strange bird in there; aside from the quarantine, they need to get used to each other.  Keets are not naturally aggressive toward their own species but it doesn't hurt to play if safe.  Also, please do not buy one off a store, do try your best to adopt instead, there are so many parakeets out there that need a good home it's a real shame to buy one from somebody who is making a living of exploiting animals.

I'm really sorry you lost your bird and I don't mean to be hard on you but the difference between wise people and just smart ones are that the wise learn from their mistakes so let's this be a lesson to you:  feed them lots of veggies and not so many seeds so they don't become overweight (veggies and fruits in the am and just a bit of seeds in the evening for dinner); establish a routine of them coming out of their cages for fly-time (make sure the room is safe for them, first), at least, three times a week; and observe them every day and if you ever see one of them not breathing right, look for other clues, like if you hear something like a clicking or ticking from their chests, that means that they have air sac mites and they need the Scatt or Ivermectin; look for labored breathing (when they do it with their beak open); puffed feathers, lethargy, nasal discharge, etc. and do get them the medicines right away.  Keets can live up to 30 years so your keet, at three, was way too young to die.