Daihinia is a network layer based on the Ad-Hoc (also called IBSS) Mode of WiFi operation, while the connection to an AP is the Infrastructure mode. These are two WiFi modes that work completely different and cannot be mixed with one another. One physial WiFi adapter cannot be connected to an Ad-Hoc and an Infrastructure network at the same time, you need two physical adapters to make such a connection. Even the new Virtual Wifi provided in Windows 7 cannot connect one virtual adapter in infrastructure mode and another one in Ad-Hoc. You still need two physical adapters.
However, if you have two physical WiFi adapters on the same computer, you can set up traffic routing between them at the IP level, in this case one adapter should be connected to the AP and another one will operate on the Daihinia mesh.
With two adapters set up as said above (one connected to Infrastructure WiFi and another one connected to your Daihinia mesh), you can share the Internet connection provided by the AP to your Daihinia mesh in the same manner as you would share an usual wired connection (see the how to share an Internet connection).