If I understand your diagram correctly, the Laptop does not have direct access to the Router, it accesses the Internet through the Desktop behind a two-level NAT (Router, then Desktop). This setup is quite strange, there is no need for an additional address translation, you can just put the Laptop in the same subnet (192.168.1.0/24) as the Router and the Desktop, and let it communicate directly with the Router, or use the Desktop as a simple router (without address translation) if the Router doesn't have wireless interface.
You will not "convert" the Laptop to IPv6, since that would hinder your ability to browse the Internet (since your router and most of the Internet still doesn't support IPv6). What you want is to install the IPv6 stack in parallel to the current IPv4 stack in both Desktop and Laptop. You run both protocol stacks side-by-side, at the same time. The IPv4 will continue to be used to access everything that is still IPv4, and the IPv6 will be used to access IPv6-enabled sites. Enabling the IPv6 stack in both your hosts will allow them to talk to each other using the IPv6 stack (as well as IPv4).
To start using IPv6 between your hosts, just go to network connections in control panel, open the properties box of your network adapter and install the TCP/IP version 6 protocol in both machines. You don't need to configure any address, they will assign IPv6 addresses themselves automatically in the fe80::/10 prefix (if there was an IPv6 router present on your network, other (routeable) addresses would also be configured automatically).
To access the IPv6 Internet is a little more complicated, since your router doesn't have IPv6 support, and I bet your ISP also still doesn't provide you direct IPv6 access. You need to transport IPv6 over your current IPv4 Internet. The easiest way to do that is to use Teredo tunneling. You will find information on how to configure that on the Microsoft website. Usually, this command is enough:
netsh interface ipv6 set teredo client
To check your IPv6 addresses:
There are other options, like using a Tunnel broker, but these are usually a little more complicated to setup.