ARP broadcasts the messages to every reachable MAC-Address, right?
ARP requests are broadcast, ARP replies are normally unicast.
So without a brute-force style scanning, I expect ARP protocol to be able to broadcast a request from my laptop, and I expect the peripheral to answer and show its IP.
That's not how ARP works. ARP resolves the MAC address for a given IP address.
For the other way around (long obsolete) RARP was defined which resolved a given MAC address to one or more IP addresses (by asking a central RARP server). Its adoption was low and it was obsoleted when BOOTP and especially DHCP became popular - you can use a reservation on those to assign the desired IP address to a given MAC address.
There's also Inverse ARP, but I don't think it's ever been used.
In a nutshell: if there's no DHCP server you can ask you have to do a brute-force scan or alternatively use a higher-layer discovery method. For instance, you could craft a UDP packet addressed to 255.255.255.255 (or directed broadcast) in a unicast Ethernet frame.
As Ron Maupin has pointed out in his comment, the best practice and future-proof (IPv6) way to discover another machine is to use IP multicast. With the corresponding network configuration, multicast may even work across routers.