You've changed the tide so that it is in the northerly direction only. It will have an easterly component as well.
Here is what I got so far:A ship is headed east @ 15 MPH.
There is a tide North-10 degrees-East @ 5 MPH.
Find the course and speed of the ship.
But the solution 15.7875 MPH is incorrect so undoubtedly my angle would come out incorrect too.
Where have I gone wrong?
You have two options.
1. You can leave the tide as it is and add it vectorially to the velocity of the boat. To do that form a parallelogram from the vectors representing the tide and the speed and direction of the boat. (It will be like your last diagram except that the line at the top will be horizontal.) Then you have to calculate the length of the long diagonal using the cosine rule, and also the angle it makes with the northern direction.
2. Alternatively, split the tide into two components 5cos(10) in a northerly direction and 5cos(80) in an easterly direction. (The two together will be equivalent to the actual tide.) That will mean that effectively the boats speed in the easterly direction will be 15+5cos(80) not the 15 that you had for your first attempt. Complete the rectangle and you can use Pythagoras to calculate the length of the diagonal.
Try doing it both ways and show that you get the same result.