You can have a look at the KEGG.db package at Bioconductor. But there isn't free (updated) access to the whole database see here)

You can get up-to-date links between kegg reaction IDs and genes using KEGG REST API. You can first find mapping from reactions to enzymes and then from enzymes to human genes:

library(KEGGREST)
rxn2enz <- keggLink("enzyme", "reaction")
enz2hsa <- keggLink("hsa", "enzyme")

Replace hsa with the organism code you need.

Check if this helps,

http://rest.kegg.jp/list/reaction

Example for batch retrieval:

A: Retriving compounds name from KEGG IDs

For anyone who might look this up, I did actually find this information within KEGG's website. The URL http://www.genome.jp/kegg-bin/get_htext?br08201.keg, which is available from the webpage http://www.genome.jp/kegg/brite.html and listed on the link for "Enzymatic reactions" shows all of the reaction information I had been looking for. This includes compound names, reaction IDs, and hierarchies. Other links from the same BRITE URL above have a conversion list of KO terms to reaction IDs and multiple other full listings that I was able to carry out network analysis around.