Our On Demand products do not use a traditional license key file. There is no license key to install.
The data for an On Demand subscription is tied to the Organization that was created when first using On Demand. As long as the subscription owner is an On Demand Administrator in that Organization, or the Subscription Owner shares their Subscriptions with that Org, the data and configuration should be retained and the product would be licensed as per that subscription.
There might not be any action required if the current subscription, as well as the new subscription (either paid or trial), have the same Subscription Owner. If not, the Subscription Owner would have to share the subscription again.
If the new subscription has already been created, please login to On Demand, go to Settings | Subscriptions and check to see what the expiration date is for the subscription. If it shows the old expiration date, please go to Shared Subscriptions, enter the email address of the Subscription Owner and click Submit. Once the Subscription Owner approves, please check again to see if the subscription status was updated.