Is there a way I can use a wireless keyboard and mouse without inserting its USB stick into the USB port?
By just using the Wi-Fi available on the laptop.
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No. A wireless mouse/keyboard does not use regular WiFi (i.e. 802.11x) and can only bind with the receiver it came with. (An exception may be the Logitech Unifying receiver, which allows connecting every Logitech device that supports it, to a single receiver - but still, it will take one USB port.)
If you do not want to use a receiver, consider using a Bluetooth mouse and/or keyboard.
I had this problem. My PC is in the office at the opposite end of the house, but sometimes I want to view the PC in the living room. The fix I found for this was to use WifiMouseHD app on the iPad to act as a keyboard and mouse over WiFi and buy a Miracast Dongle to copy the screen over WiFi. Not sure if this will fix your problem. app can be free, but I paid a small amount for full version, and the dongle was about £30 I think. I'd be interested to know if you found an alternative solution. Good luck!!
To fill in a bit more explanation to some of the great answers already given:
The WiFi (802.11 a/b/g/n) requires having some kind of a central authority to pass data around -- called an
Your computer connects to an access point as a client... often having to enter some kind of authentication -- WEP key/WPA Key/etc.
Imagine if you had to enter this key into your keyboard or your mouse before they could connect to the access point and then connect to your computer.
In reality, this just doesn't work -- so alternative methods of connecting from device to computer were invented. Simpler protocols such as bluetooth and whatever else is out there.
 For the moment, I am skipping adhoc networks, but those are rare.
Yes, this is possible, in theory, but the chances of it being possible with the keyboard you already have, or with any keyboard that is likely to exist COTS, are pretty much zero.
Data radios come in many different types and variations, and they are not interchangeable. For this to work, would require that the radio built into the keyboard be built to operate on the Wifi band, using the Wifi protocols. In practice, people who design and make wireless keyboards do not usually choose this band or this protocol, for a variety of technical reasons, most notably that Wifi radios tend to be very power-hungry, negatively impacting battery life.
Most wireless keyboards instead use a Bluetooth radio, with a significant minority using some other band that I don't know the identity of, but do know that it is neither Bluetooth nor Wifi. The closest things I know of to any existent Wifi-using wireless keyboard are a few tablet/smartphone apps which turn those devices into wireless keyboards, and which use Wifi because the radio is already there in pretty much all such devices.
Although possible in theory, you would find that it would drain the battery of the mouse very quickly (that is, in a matter of days). Wi-Fi is extremely power hungry, and low power solutions such as Bluetooth or ZigBee are far better suited.
In addition, Wi-Fi is difficult to set up without some sort of visible console to select the Wi-Fi password, encryption type, etc., which creates a bit of a Catch-22 - how do you enter the password for Wi-Fi if you don't have a keyboard set up yet?
Thus, the market has moved towards efficient, low-power, low range devices to transmitting keystrokes and mouse movements to a laptop.
This example happened to me with a user on a Windows 10 Dell Z.. model PC:
User reports that her Microsoft wireless mouse is working but her wireless keyboard stopped working.
*Reviewed user's desktop computer and could not find a USB receiver for the wireless mouse and keyboard set. A fellow colleague of the user has the same Microsoft wireless mouse and keyboard set and uses a USB wireless receiver.
*Reviewed Device Manager and discovered a Generic USB Hub error (!).
*Logged in with Local Admin account and selected the Device Manager option to “Scanned for Changes”, and there was no status change for the Generic USB Hub.
*Confirmed that the latest drivers are installed.
*Restarted user's computer and now both the Microsoft wireless mouse and keyboard connects and works properly.
Notes: 1. There is no USB wireless receiver connected to the users desktop computer. 2. Devices and Printers now shows two Microsoft Wireless Transceiver entries. 3. Device manager now shows two Wireless Keyboard Filter Devices and two HD-compliant mouse entries (see attached images 1 and's 2). 4. Also,there is no make, model or serial number tags/barcodes on neither the mouse or keyboard.