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 for 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).
*The Headspace + Spotify Premium bundled subscription is currently only available in: Sweden, Denmark, Norway, Finland, Netherlands and the UK.