You can do that by using a conditional card. Then you need to wrap the mini media player card inside the conditional card. For instance:
- card:
artwork: full-cover
entity: media_player.speaker
group: false
hide:
controls: false
power: false
progress: false
volume: true
source: true
icon: 'mdi:speaker'
info: scroll
style: ''
tap_action:
action: more-info
type: 'custom:mini-media-player'
conditions:
- entity: media_player.speaker
state: playing
type: conditional
The condition looks at the state of the entity and only shows it only when the condition is met. You can also use the multiple state_not variables if you want it to show not only when playing, but also when paused. If you chang the conditions to state_not: off it basically says show it always except when state is not. You can add multiple state_not variables.
In this case I advise you to not use state: playing but use state_not: off, otherwise you will only see the card when playing and it will disappear when you pause it 
I advise to read the documents I linked about the conditional card if you have any issues.