Go to Toyohashi and take the shinkansen. Not taking the shinkansen ensures you spend 3 or 4 times as long, change 3 or 4 or more times, ride much less comfortably, and save about ¥4,000.

www.hyperdia.com

1) 5480 JPY, 6 hours, Meitetsu train to Toyohashi, then JR Tokaido Line with 3 train changes

2) 9080 JPY, 2 to 3 hours, Meitetsu to Toyohashi, then shinkansen. From Toyohashi there are Hikari shinkansen at 6:40, and then 8:45 and every two hours after that until 20:45.

10:17 Higashi Okazaki to Toyohashi

here is good itin:
10:17 Higashi Okazaki to Toyohashi 10:37
10:45 Toyohashi to Tokyo 12:10

that is 8730 unreserved and just under 2 hours

There are Kodama shinkansen between those Hikari from Toyohashi, but they make 9 stops vs 2 stops on Hikari.

Might be worth the 510 yen to get a reserved seat on the shinkansen as it might be busy on the holiday.

The shinkansen is the only direct train. If you want to spend 4 hours to save 3250 to 4000 yen then the connections on the Tokaido Line might not be difficult as they very well could be on the same platform.

In one hyperdia example connections between Meitetus and JR in Toyohashi is 8 min. Then 12 and 21 minutes connection times - enough time to change platforms if you had to. One connection in Atami is 3 minutes - this probably means following everyone across the platform to the other train.