Lux's E does not do damage until it is activated again or times out, Rylai's procs only on magic damage, so the slow from Lux's E can't proc Rylai's passive.
Slows in LoL other than just item slows can stack, so any slow applied after a target was already slowed increases the slow amount. In patch 0.9.22.15 "Removed the internal 6 second cooldown." for Rylai's passive. So I'm guessing the damage from Lux's passive and the damage from Lux's E can both apply Rylai's passive, but since item slows don't stack it will only refresh the slow, not increase it.
Also note that Lux's passive is single target and will therefore apply Rylai's 35% slow, while the damage from Lux's E is aoe and only applies the 15% slow. Any item slow on a champion already affected by an item slow will only apply if it is a stronger slow than the previous slow. Keep in mind that proccing the passive after activating Lux's E will increase the slow as well as refresh it.
Lux E description from lol wiki:
ACTIVE: Lux sends an anomaly of twisted light to a target area,
revealing it and slowing enemies within 300 range for up to 5 seconds.
At the end of the duration or if Lucent Singularity is activated again,
the zone will detonate and deal magic damage to enemies in the area.
Rylai's passive from lol wiki:
Unique: Dealing spell damage slows the target's movement speed by 35%
for 1.5 seconds (15% for multi-target and damage-over-time spells).
Notes on item slow, in Rylai's lol wiki page:
The slows from items will not stack with each other.
(i.e. Rylai's Crystal Scepter does not stack with Frozen Mallet
The strongest item slow will be effective until it is expired.
Any other item slow that was applied and not expired will take effect after that.