Need help choosing my first amp for my first speakers

One question, is there a reason why you are calculating for 8 ohms when the spec is Nom. impedance: 4 ohms?

Personally I’d want at least 200Wpc at 4 Ohms to handle the W5’s impedance dip(s), and to get that you’re gonna need to look at amps that are at least 100Wpc @ 8 Ohms as that’s the number you’ll more readily see without having to dig deeper into the specs. 
 

Here’s another all-in-one idea that if you can stretch for would likely serve you better than the Wiim Amp as it likely has a much better DAC and power supply and probably sounds better overall.  The USB input is limited to 24/96 but that should be more than fine for now.  Just another option FWIW.

https://www.usaudiomart.com/details/649987921-bel-canto-c5i-integrated-amplifier/

How big is your room, how loud do you want to play?

Those should have some excellent midrange clarity and sound stage, but aren’t likely to "rock the house" as-is....I’d want to put a nice tube amp on them to get the most from what they do best, then run a well matched active subwoofer to help where they don’t excel. .I’d give you’re current amp a whirl and see how it goes, but would likely consider the subwoofer even if I kept the amp. Your room will definitely come into the equation though.

@jcjcjcjcjcjcjcjcjcjcjcjc 

Check out the Pine Tree integrated amp on US Audio Mart, slightly above your budget at $ 875 but would be an Integrated that will have enough clean power to really drive your speakers and isn’t one you’d have to upgrade too quickly, you could build off of it and upgrade if you really got the itch once you had everything else built out.  Has a DAC built in that is reviewed well, all you need is to provide a source.  Maybe the seller will give you a first time Audiophile discount. I tried posting the link but the site was blocking me from doing so. 
 

Cables,  don’t roll with Amazon, look at Pine Tree Audio, great, hand made cables that won’t break the bank but won’t need to be upgraded.  You can also find some EBay sellers that make their own cables, you can get some really well made cables made with quality materials for a really great price but you have to sift through and find the right vendors.

When you chose an amp / integrated, you’ll need to be selective as your speakers are harder to drive, it’s not all about watts but generally an amp that doubles Watts from 8 to 4 ohms and is rated to drive speakers throughout the range, your speakers dip, is key to letting your speakers shine and not burning out whatever amp you roll with. 
 

Good Luck. 

KnotScott asks good questions, but you have set a high bar for yourself by choosing such inefficient speakers. I would start with a Fosi ZA3 amp, which will give you a reliable clean power supply and can be bridged if more output is desired later. A WiiM Pro Plus will serve you well as a front end. World’s Best Cables are a lower priced equivalent to Blue Jeans, and in my experience (mostly with their Mogami interconnects) are dependably high performance. Get one of these too.

This solution will complement your diminutive speakers by being a tidy compact system of 2 components and a power brick.  Don't forget to get speaker stands as well.

I recommend getting The complete guide to high end audio by Robert Harley. An excellent source for learning about this great hobby.