You could continue the binary search but that's not relevant anymore - the problem is to minimize the worst case, and your worst case is 51 while there exists a solution whose worst case is 14.

You keep using it. Binary search is the right way of thinking about it, but in some sense you haven't really created an even split by dropping the egg from floor 50 - because you know that you can figure out the correct floor faster using two eggs than one, you could check floors 51-100 much faster than you could check floors 1-50 (Even if you just did the naive binary search approach you first applied - i.e. if the first egg didn't break on floor 50 then drop it from floor 75, and then after that use the second egg to check each floor, it would only take you <=25 goes to find the right floor in 51-100, wheras it would take you up to 50 goes to find the right floor in 1-50).

So the correct floor at which to make the first drop is clearly not floor 50. If you dropped the first egg from floor 40 you would do better - it'd take you up to 40 goes to find the right floor if it's below floor 40, and up to 31 or so goes to find the right floor if it's above (because you can reuse the first egg to check whether it's in 40-70 or 70-100). Floor 40 is in some sense a "more binary" place to drop the first egg from than floor 50.

If you keep following this line of thought, and remember that whenever you make your approach more efficient, you can recursively apply that more efficient approach to "what do I do if the first egg doesn't break", you'll arrive at the solution given in the article.