Private duty is unlikely to require board oversight. After all, you don't even need a license to perform personal duties privately. However as an LVN, you would have to stay within the scope of your license so a careful reading of the practice act would be in order - you may even have to work under an RN developed care plan, who may have to work under a physician's orders.
You will also need to develop a good contract for your clients. Effectively, you will be operating a private duty agency. See if one of your patients will give you a copy of their current agency contract. That will serve as an excellent template for your own contract, and give you a heads up on any missed regulatory requirements. You may not be required to carry insurance by the state, but you really do need to carry at least a mil of professional liability and a similar amount of general liability. Anything that happens in the home you are working in such as a fire could prove devastating to your personal finances if you are not adequately insured.
In most states your professional license is all you need to do business. However, in some states, personal services are subject to sales tax. If that is the case in your state, you will need a business license to collect and pay sales tax. Most business questions will be met by any book on starting a business (like whether you should incorporate or not), and there may be a specific book on starting a business in MN - there are for most states. Browse a bookstore or Amazon.
Good luck! Don't sell yourself cheaper than established agencies. I usually end up with the highest bill rate because I am marketing only me, not some unknown quantity. Mind you, I'm a travel RN, but there is a close parallel here.