I think this would be easier to use improper fractions since you are going to have to multiply. Also, since 2-11/63 is NOT a reasonable width, possibly the area of the kitchen is 1.75 the area of the room (not 1 minus 3/4 = 1/4). So here goes:
The area of the room is 99-3/7 = 696/7. The area of the kitchen is 1-3/4=7/4 of the area of the room (WHICH IS MORE). Therefore, the area of the kitchen is 696/7*7/4 = 696/4 (since the 7/7 reduces to 1) which equal 174 square feet for the kitchen area.
If the length of the kitchen is 14-1/2 = 29/2 feet, then solving Area = Length * Width...
174 = 29/2 * W ==> multiply both sides by 2 and divide both sides by 29 to clear the multiplier of the variable [you could have chosen to set this up as a proportion to solve the equation 174/1 = 29W/2]
2/29(174/1) = 2/29 (29/2 * W)
348/29 = W ==> multiply across the numerators and across the denominators
12 = W ==> division
Therefore the Width of the Kitchen is 12 feet.
NOTE - the Area of the Kitchen is 14.5 ft X 12 ft = 174 ft2 which is clearly MORE than the area of the room (since it is 1.75 times it)