Well, there are several open-source libraries already available that you can choose from depending on your preferred programming language. Some of the popular ones include c-lightning, LND and Electrum server.

If you prefer Python, PyNostr is a Python library for Nostr released under the MIT License and it provides simple calls for generating transactions to relay via signed Nostr notes submitted over Twitter using Lightning Network payments with python.

As for sending LN invoices via DMs, Flask-Pynostr is a micro-framework which can be used as starter code since it helps in integrating the payment mechanism to a custom flask app. WIth these tools you might achieve just what you intend.