The traditional way (even before people understood how the genes work) is to just breed from the most completely white birds in each generation.
The Ii birds (heterozygous Dominant White) are more likely to have bits of black leaking through, and the II homozygous birds are more likely to be completely free of black. If you want to be sure, you can test-mate individual birds to ii birds (Light Sussex father or Rhode Island red would be suitable mates for testing.) If a given bird produces any chicks that show i then of course they have i, but producing 6-8 chicks that all have I is a pretty good indicator of being homozygous for I. (If you're not familiar with calculating probabilities, the chance of getting that many chicks with I from an Ii bird is 1 in 128 for 6 chicks, 1 in 256 for 7 chicks, 1 in 512 for 8 chicks. The more chicks you hatch, the more sure you can be.)
For the red leakage, if you have any birds with no red leakage, you could obviously choose to breed from them. Just hatching a large number of chicks in your second generation might yield some with no red leakage.
If your original Light Sussex rooster does not show leakage, you could cross daughters back to him, and pick the offspring (both sexes) that have no red leakage. That cross could potentially give you some Ii birds with no red leakage (might show bits of black leakage), and then you would breed those together to get some II birds (hopefully with no red leakage or black leakage.)
Many breeds have the main color genes that we know about, plus some other genes that are not named or properly understood that modify the expression of the main ones. Crossing back to either parent breed could pick up more of those unknown modifiers (probably more useful on the Light Sussex side than the Sexlink side, but might be useful either way.)