See this link for the officially published regulations from the Government Printing Office (now online too). http://www.ecfr.gov/cgi-bin/text-idx?c=ecfr&rgn=div5&view=text&node=14:126.96.36.199.15&idno=14
Subpart A starts out with the applicability of "this part" and then states it is applicable to kites weighing more than 5 lbs intended to be flown from a cable or rope. So this subpart applies only to kites weighing more than 5lbs. There is the further applicability restriction of the "cable or rope" flight - but because we all say flying line (not string) and kites over 5lbs are probably on 500lb line or better - I would not want to be arguing over whether my 1200 lb dyneema line is a rope or not - I think a reasonable tryer of fact would find that to be a rope.
What does Subpart A do with respect to kites weighing more than 5lbs? 101.5 prohibits their flight in prohibited or restricted areas unless permission is sought and obtained. 101.7 prohibits their use anywhere if the present a hazard.
Although it would seem like these requirements should apply to any kite anywhere (and that is written on a number of kite sites), the regulation does not have that effect. For those that have been there, think about kite flying on the US National Mall around the Washington Monument in Washington, D.C. All of that area is restricted airspace because of the Whitehouse, Congress, etc. Yet no permission to fly a kite is required. And the only restriction on kite flying on the mall is no manja allowed - and that's a Park Service rule.
Subpart B is a different part. So the applicablity provision of A do not apply, and indeed B has its own applicability provision. It applies to "kites" - no definition given - and no weight limit. So comparing and contrasting to A - Subpart B applies to kites of any weight.
What does Subpart B do with respect to any kite?
(1) No flight more than 150 feet above the surface of the earth unless, at least 24 hours before beginning the flight notice is given to the FAA
- where notice is given, flights between sunrise and sunset require lines have colored pennants or streamers attached at not more than 50 foot intervals beginning at 150 feet above the surface of the earth and visible for at least one mile
(2) No flight less than 500 feet from the base of any cloud;
(3) No flight more than 500 feet above the surface of the earth;
(4) No flight from an area where the ground visibility is less than three
(5) No flight within five miles of the boundary of any airport.
So 500 feet is the limit - but only where notice is given to the FAA. Note that this is just notice -- you do not have to get an approval. The FAA uses the information to publish an Airman's Notice warning planes about the activity - well they are supposed to but I have not found many examples of that on the internet - perhaps because there aren't many notices given.
The regulations do have an exception for giving notice - if you are flying within 250 feet of a building (i.e. sheilded structure) then no matter how high the building is you can fly your kite if you stay below the rooftop and do not obscure any airplane lighting. No guarantee on what winds you will get - or what the owner of the building may have to say about the flying.