I've found a third possibility - a book in the Questers series by E.W. Hildick.
From the Goodreads page for the first book:
Peter, who is bedridden, has a telephone and a remote-control television unit on his bedside table—and if the Questers’ Master Plan works, he’ll have a walkie-talkie set that will keep him in touch with his friends’ activities. Andy and Julius are the full members of the Questers, and Peter’s sisters, Eve and Ruth, are associate and probationary members. By combining forces they hope to win the talent contest, the pet show, the ice-cream eating contest, and the all-important treasure hunt.
So far, we have a bed-bound kid, and his friends who come to visit him. We also have a TV screen, which could be the computer screen you mentioned, and it's remote controlled.
The book I have in mind would be the third one in the series.
Note that electronic equipment, considered to be special in some way, is on loan to them from a dedicated company - thus increasing the chance that subjects like the knapsack cipher would come up.
Here are some pictures from the interior - does the art match the style of the line art you remember?
I can find almost no information on the second book in the series, but here's its cover: