The conditions shown before you commit to any streaming service subscription I've ever seen clearly state the fee is non-refundable once paid. If you subsequently cancel while in the subscription period, that prevents the subscription from auto-renewing when the subscription period ends.
If you subscribed through Roku instead of directly with the service you're subscribing to, Roku just acts as an agent and has to pay that service.
If you can't trust your kid to read the conditions and abide by them, you can log into your account at my.roku.com and set up a PIN for your account and set the conditions under which the PIN must be entered, such as when making a purchase.
Roku Streaming Stick +, 3810X, Ser YH0059427035, wifi LivRm - Samsung UN55ES6100
Roku Streaming Stick, 3600X, Ser 5S56D8240827, wifi BedRm - JVC EM37T
Roku 3, 4200X, Ser 4124C7087583, wifi GstRm - Sanyo DP26640
For account and billing related issues, please reach out to our support team directly here for your 'Account or billing' issue, here: https://support.roku.com/contactus. They will be able further assist you.