Headspace can only be accessed through the Headspace App or from a desktop computer. Unfortunately, you wouldn’t be able to listen to Headspace content via Spotify.
The Headspace + Spotify bundled subscription* allows you to subscribe to both Headspace and Spotify in one subscription (vs. two!). So, all you’ll need to do is create an account for each (if you haven’t already) and register for the bundle.
*The Headspace + Spotify Premium bundled subscription is currently only available in: Sweden, Denmark, Norway, Finland, Netherlands and the UK.