I'm having trouble finding anything that actually has all of the ports and connectors that (I think) you'll need, but I can explain how I would go about this. If you can get me specifics on what display outputs are available on your PC and laptop, as well as your monitor resolution and refresh rate, I'll take some time to find you the exact equipment.
What you want is a KVM switch, a mechanism designed to use one keyboard, mouse, and monitor (in your case two monitors) as a workstation that can dynamically swap between two computers. I recommend this over a standard dock for ease of use, flexible configurations, and preservation of desk surface real estate.
You've basically got two approaches. One is expensive and the other is somewhat convoluted.
That site is a good place to start, they make high end switches and they're just about the only company I can find that even does digital video formats or dual monitor outputs in any significant capacity. Almost everything else I've been able to pull up has been either analog (VGA) or DVI with maybe a few DVI-Dual Links in there. The downside of this approach is that you're probably going to drop $200+ on this solution.
The other method you could use would be to downgrade your cable connections to DVI (or whatever you can find that is compatible) and get two cheap units that handle single monitor outputs. This is still likely to run you $60+. There's also a concern about what resolution and refresh rate you're running at, as each type of cable has a ceiling on how much data it can transfer and thus what resolutions and frame rates it is capable of transmitting. I haven't found any single monitor HDMI or DisplayPort boxes that were cheap enough to consider over the StarTech products linked above, but you may have more luck. I searched Amazon, Newegg, and PCPartPicker if you were curious. Office Depot or Office Max might be worth a try.
PLEASE KEEP IN MIND: Most of these KVM switches do not come with extra cables, so plan your layout and figure out the types and lengths of cables you will need beforehand and order them alongside your KVM solution. This will save you headaches and irritation and time waiting for shipping.
Again, if you want to reply with those related specs that I mentioned above, I can find some time before Friday to help you hunt down specific hardware.
Keyboards are really up to you. If you don't mind a 60% board I can recommend an Anne Pro, I have one for work and love it; however, I haven't used it for gaming since it has a tiny layout and red switches. I can confirm though that it has built-in bluetooth (comes with dongle for non-BT devices) and hotkey device swapping for up to four connected devices, so it would fit your requirements there. Bonus points for being under $100 and having quality RGB because colors are fun. If that doesn't catch your eye, I know that Filco also has some good offerings, and CODE keyboards are always worth a look as well.
The subreddit for mechanical keyboards has this wiki page listing wireless options:
You could also search around on that forum to see what other people have used and what they liked/disliked, it's an active community with a good amount of content.