Streaming virgin

If your goal is simply "discovery", an Auris BluME HD bluetooth receiver will do the trick at just over $100. It is bluetooth and you are using Spotify, so it will be a lossy digital stream and sound quality will not be optimal. However, I use one and I can tell you it is plenty good enough for me in my system. I use it regularly for the exact application you mention - discovery. For me, Spotify Premium, and their "Discover Weekly" playlist, have resulted in more "new to me" music than any other source. My second most successful source for new music has been streaming Radio Paradise's internet radio, which you can also do with the Auris BlueME HD.

To use the above you just need to plug the analog output of the Auris BlueME HD into a spare line-level input on your integrated amp. From there, pair any bluetooth enabled device to the Auris, go the the app of choice (Spotify, Radio Paradise, etc.) and play. It will transmit from your device (Kindle, mobile phone, etc.). Lastly, if your integrated amp has the DAC module in it, you can connect the Auris using a digital cable and use the DAC in your integrated amp. BTW, I do this in my system and there is a marginal improvement in sound quality using the external DAC vs the internal DAC in the Auris.

Easy, Peasy, only $100-ish dollars and works great. Likely won't match the SQ of your CD setup, but perfectly fine for discovery and non-critical listening.

@dagery Does the kindle have digital out? If it does you could go digital out to the USB B input on your integrated amp.

If not, you could purchase an Apple iPad (or any pad that has digital out ) and use the Spotify app and go digital out from the PAD to the USB input on the integrated. This will be a good way to get your feet wet without spending to much. I have an iPAD that I do this with using the Qobuz app on the iPAD. If you buy the iPAD air I think it has a USB C connector instead of lighting connector. USB C to USB B connectors are easy to acquire. My iPAD has the lighting port. I just bought an adapter that goes from lighting to USB A then has a USB B termination that will go into my DAC.
The only negative is the PAD is hard wired to the integrated and runs on battery during this time. The advantage is that if you do decide to invest in a dedicated streamer you can always use the iPAD as a controller.

The other advantage is if Spotify lets you download tracks.  Qobuz has a feature where in you can download HiRez files local do the Tablet/PAD and then play them.  In fact its the required method to play anything higher than Redbook CD quality.  Can come in handy if you have planned network outage or trouble shooting network SQ issues.

Example
AudioQuest USB C to USB Bhttps://www.audioquest.com/cables/digital-cables/usb-b-to-usb-c/usb-b-to-c-forest

That integrated has an 1/8 inch mini jack input on the front of it. Have you tried using that with the kindle? you mentioned you were using earphones. Of course this will be using the crappy DAC in the kindle.

Is there a DAC within the integrated amp or transport?

You may already have a DAC and only need to have a streamer to your system.  

@mrklas
Yes that Amp should have a ESS9018K2M according to the user manual.  That is why I was recommending going digital in.

Then as a subsequent step you could fire up a trial subscription of  Tidal or Qobuz and try hi rez to see if its worth the spend. Deezer is also an option and streams at RedBook CD quality and has a vast music collection.