Summary: the NAT gateway must be in a public subnet, and it must have a route to an internet gateway. AWS documentation covers this here and here.
Look that nat gateway is in a subnet that is accessed through a normal internet gateway in your case both Host1 and Nat Gateway can be in the same subnet or in a subnet that forwards the traffic through a normal internet gateway.
In a simple image what you must achieve is the following:
So consider Nat Gateway as another host that forwards traffic to a subnet that has internet gateway. The diagram above shows how each machine should be connected to internet. In your case
Host 2 accesses
Subnet 2 and through Nat Gatewaty and subnet 1 via a normal Internet gateway accesses the Internet.
In your case, either the subnets
subnet-065ae3de09e9f8355 are suitable to host the Nat Gateway. But Nat Gateway must NOT be in the subnet
In order to change the subnet of the Nat Gateway you must create a new one Nat Gateway afterwards update the route table to utilize the Nat gateway. If you want to keep the Elastic IP wait for the old Nat Gateway to be deleted. Afterwards, use the new one.
And remember once you are creating the new Nat Gateway in this field as shown bellow:
Select either the subnets