With a cooler mash tun you're going to lose about .2 gallons per pound of grain to grain absorption. So 13 * .2 = 2.6 gallons. I have no idea what your boil-off rate is, but mine is about 1 gallon per hour. Assuming that, you'd add another 1.5 gallons on top to account for evaporation. So 5 + 2.6 + 1.5 = 9.1 gallons total water needed.
If you are mashing in at 1.5 quarts per pound, that would give you 13 * 1.5 = 19.5 quarts which is just about 5 gallons of strike water. The rest would then be sparge water (about 4.1 gallons).
That's using rough numbers for my system with my igloo ice cube cooler, which I tilt a bit to aid in getting more liquid out during sparge, and my Blichmann 15 gallon pot for the boil kettle. Your system might be different, so it would behoove you to figure this stuff out a bit so you can make adjustments on the fly. e.g. if you are over your estimated volume you can boil a bit longer, or if you are under you can add a bit more water to the kettle after the sparge. However my suggestion with the sparge in particular is to let your MLT continue dripping into an extra bucket after you finish your sparge. You will then be able to heat up to your boil a bit faster, while still getting the extra wort that you'd otherwise be leaving in the tun...then when the boil is getting close to a start, you can take the contents of the bucket and add it to the kettle. Usually for me this is between 1-2 quarts of extra runoff, depending upon how impatient I am with the initial sparge.