Red-Black tree height from CLRS
step 1 : At first let me say that , property 4 which states that children of a red node , should be black , comes from the definition of red-black tree , because 2 (or more) red nodes can't come after each other .
step 2 : For the statement that is a result of property 4 :
let's consider a path from root to a leaf , which consists of 2K nodes, and the number of Red ones, is more than the number of black ones, so at least number of red nodes must be K+1 and therefore number of black nodes is K-1. From property 4 we shouldn't put 2 (or more ) red nodes consecutively , so between every 2 red nodes , we should put a black one and since we have (at least) K+1 red node and (at most) K-1 black nodes , after we put the Kth red node , we will run out of black nodes and we have to put the (K+1)th red node right after the Kth red node , so we get two red nodes placed consecutively (you can prove it by induction )and this is in contradiction with Red-Black tree definition. so "at least half the nodes on any simple path from the root to a leaf, not including the root, must be black" 
step 3 : For the Height of the tree : 
we know that the height of a tree ,  is the number of edges in the longest path from root to a leaf and from what we proved in step 2 , at least half of the nodes on this path can be black , so if we have the height of h , at least we have h/2 black nodes , which comes to say that the black-height is at least h/2 . 
step 4 : And for your solution , I guess you are being too general , in my idea you'd better consider number of red and black nodes , in an arbitrary path from root to a leaf rather than in the whole tree .
Hope it works as an idea not the whole solution :)