Sekaijin got right to the point. Your proposed route is bad. Avoid the Badger Rd area - it's industrial, and not bike nor people friendly.
Are you heading to Parkwood Hills neighborhood if I may ask? (The streets named after National Parks)
You could pick up Cap City at Nob Hill and go left instead of right, and then follow it to the Verona Rd underpass (10-12 miles or so?), and it would be nice and scenic on the path but I find that a little confusing once you get to the underpass. It generally involves a lot of turns, hard to follow bike paths and stop signs once you get on Williamsburg. However, you could also stay on Cap City just as far as on Seminole, turn right and take the rest of the route from there.
The only reason why I know about the Odana Rd path it is because I spotted it one day while at St. Vincent's and later decided to figure out where it started and ended. It is a little hard to find from the east, and a little hard to get out of from the west.
iab if the timing is good us westsiders could just lead you on a ride to your destination.