day trip from London to Brussels?

It takes two hours by train. The highest speed train Eurostar (the only way I would go there on day trip) list the travel time at 1:22. Whether this is worth the travel time all depends on the make up of the person.
The best thing you can do is read or Google a travel guide on Brussels or ask the companion to inform you so you can see if there is anything there of interest to you. If not just let the travel companion go on the day trip. I personally like visiting the Grand Place in Brussels and the beer in Belgium is very special in world. But not so much that I would spend more or less $200 to get there and back on a day trip. I can't afford it.

If by doable you mean physically possible, then yes it is. However, and this is just my personal opinion, it's not worth the time and trouble. I have nothing against Brussels, I'm one of a minority on here who likes Brussels. But, considering that it takes at least 3 hours each way, whether you fly or take the Eurostar train, that doesn't leave much of the day to enjoy. I think the time would be much better spent enjoying more of London or taking a day trip to someplace closer to London.

EDIT: Just a reminder about the train. While the actual time from station to station is just over 2 hrs, you still have to take into account getting to and from your hotel/apt and St Pancras station and the fact that you have to check in at least 1/2 hour before departure (on both ends) and that adds quite a bit of time. The same goes for flying. The actual flight is short but there's still getting to and from airports and airport security and boarding times.

If you're after a taste of Belgium, consider smaller towns, especially if just for a day. We have visited Bruges, Ghent, Tournay and Kortrijk, all lovely and possibly preferable to the huge Brussels.

I wouldn't cover Brussels as a day trip from London if you only have 5 days there (3 full days if you have 4 nights). I consider Brussels to be the most disappointing capital city in Europe too!

Belgium is an hour ahead, so you lose an hour getting there. Door to door will be at least 4-4.5 hours in each direction unless you are staying in the St Pancras area, as the recommendation these days is to be at the station an hour ahead of the train. You wouldn't have much time to see the sights of Brussels before having to return.

Sorry, but this is not a good idea at all.
First, way too much travel time and expense.
Second, Brussels, except for the Grand Plaza is not a top city, compared to what you could find on a day trip within England.

As you said, "there's so much to see in London." And Brussels is not anyone's favorite around here, even though I'm sure you'd enjoy a stop when it was more convenient. If he insists, I'd let him go alone. Eurostar tickets are cheaper when reserved well in advance (weeks or even several months). A nearby daytrip within England, if you need one at all, would be cheaper and more flexible.

Brussels is a bad place to pick as a day trip as the interesting sites are very spread around.

Don't overlook Paris as a day trip via Eurostar. Take an early train over and a late train back. You can take food and drink on board plus snoozing is quite refreshing, speaking from experience, both going and returning. Tickets go on sale 180 days prior to your travel date at Eurostar.com and get progressively more expensive to nearly outrageous the longer you wait.

You have a very short time in London. As Jennifer said, 3 full days if you are spending 4 nights in London. You may have part of a day on the day you arrive (perhaps a half of a day), and part of a day on the day of your departure.

You will barely have time to get a quick taste of London, not even to experience it fully, and all it has to offer.

I would save Brussels for another trip and say no to a day trip from London to Brussels.

If you do not know how very much there is to do in London--or in England as day trips out from London--I strongly suggest that you buy the Rick Steves London guidebook and read up on all there is to do.

Don't forget the one-hour time difference between Brussels and London also eats up more of your day and means your return trip takes "longer."

Indeed, you'll barely be scratching the surface in London in five days. Sooooo much to see and do.

Edit: oops, I see Jennifer mentioned the time difference.

There are plenty of day trips in England that are way better than Brussels. I doubt that anyone on their deathbed says, "If only I'd been able to see Brussels"!