If by "any sort of cool down items" you mean " 34% worth of cooldown reduction" then yes, it will be constantly available. In the early game, it's far more likely for her to only have 10-20% CDR.
Her ability to keep her AoE up constantly with enough CDR is intentional.
She is intended to act as a psuedo-ranged carry, you should treat her as such, much as you would treat a Teemo for example.
Having a kit that works with itself is not a sign of being overpowered. For example, Singed is not overpowered just because his flip and glue help get and keep people into his poison trail, nor that his ult helps him close distances and be able to survive in the midst of a team-fight.
Annie's ability to consistently last hit with Q or "spam" (within a loose definition of spam) her E to build up charges is also not something that makes her inherently overpowered any more than Veigar's Event Horizon enabling the otherwise hard-to-hit Dark Matter does.
Here's the questions you have to answer. Does Kayle hit harder than she should? If so, should that damage be reduced, or should her utility be reduced to compensate? Is it even a matter of numbers rather than your own skill or approach to dealing with the champion? Have you looked at different champions that can be good at countering Kayle, and are they reasonable counters?
Personally, I don't think she's problematic at all, her damage output is definitely lower than true carries like Vayne, Tristana, or Varus. Her utility is good with her ult, and her heal is efficient, but the heal isn't strong in and of itself, and the ult can be worked around with CC, disengaging briefly, or switching focus targets. But that's just my view.
In the mean time, if it makes you feel any better the removal of MP/5 on Nashor's Tooth is something of a minor nerf to Kayle's early game sustain.