I recommend you would leave Kyoto City in the afternoon of July 7 and make a Nikko roundtrip later from Tokyo.
(Generally speaking, hotels in Japan accept changing dates without a handling fee until at least 3 days before the earlier check-in date.)
I suppose two days (while you would be still tired) may not be enough to tour Kyoto City unless you have visited the city and got accustomed to transportation options there.
And, Tokyo is much closer to Nikko than Kyoto, in both distance and time.
Note: Kansai and Kanto Regions will still be in a rainy season during your journey.
Which area in Osaka / Kyoto / Tokyo will you stay in?
Is it worth buying the pass? If you fly out from Haneda Airport [HND] or Narita Airport [NRT], you will not need any JR pass.
To travel from KIX to Osaka City, you can use either a ticket (1030 yen) in a one-way ICOCA & Haruka package, which is to be used on JR lines, or a Kanku Chikatoku Ticket (1000 yen), which is to be used on Nankai + Osaka Municipal Subway lines with a transfer in Namba.
Note
: JR Ltd. Express "Haruka" trains do not run to Osaka Station in Umeda area. Stations in Osaka City for "Haruka" trains are Tennoji and Shin-Osaka.
: Nankai trains from KIX run to Namba Station, not to Umeda Station.
Your options to travel from Kyoto City to Tokyo Metropolis are explained in this article.
http://www.japan-guide.com/e/e2363.htmlThe easiest and fast way is to take a "Nozomi" train on JR Tokaido Shinkansen from Kyoto Station to either Shinagawa Station or Tokyo Station.
To travel by JR from Kyoto Station to a station in Tokyo, usually you get issued two JR tickets: a ticket for the whole way (to your JR station in Tokyo) and a Shinkansen ticket.
If you do not visit Kinugawa Onsen area in Nikko, a Marugoto Nikko Tobu Free Pass is recommendable; you can choose a Tobu station in Tokyo where you start using this pass.
If you visit also Kinugawa area in Nikko in addition to Lake Chuzenji area and/or Yumoto Onsen area, a Tobu All Nikko Pass is recommendable; you should start using this pass from Asakusa Station of Tobu.
http://www.japan-guide.com/e/e2358_002.html