You may use Debian 8.11 "Jessie" , Debian 9.13 "Stretch" , Debian 10.13 "Buster" , Debian 11.11 "Bullseye" or Debian 12.8 "Bookworm" right on your Keenetic router.
http://files.keenopt.ru/ is a good example of working Debian on Keenetic Giga III.
! Dropping mips architecture
! Debian 12.3 image release delayed
Requirements
USB-port on router,
Ext2/Ext3/Ext4 formatted flash drive or HDD. You may use EaseUS Partition Master free on Windows to format USB drive.
Installation
Connect USB drive and get access to it via FTP or SAMBA,
Make install folder on USB drive,
Put debian-jessie-8_11-mipsel.tar.gz or debian-stretch-9_13-mipsel.tar.gz or debian-buster-10.13-mipsel.tar.gz or debian-bullseye-11.11-mipsel.tar.gz or debian-bookworm-12.8-mipsel.tar.gz file to Install folder,
( for KN-2011 or KN-2111 - debian-jessie-8_11-mips.tar.gz or debian-stretch-9_13-mips.tar.gz or debian-buster-10.13-mips.tar.gz ),
( for KN-2710 - debian-buster-10.13-aarch64.tar.gz or debian-bullseye-11.11-aarch64.tar.gz or debian-bookworm-12.8-aarch64.tar.gz)
Open OPKG page, pick your USB drive and hit "Apply".
Give your Keenetic a couple of minutes to unpack necessary files, you''ll see following syslog messages if all went fine:
[I] Jul 19 13:24:28 ndm: Opkg::Manager: disk is set to: fb4e34d1-3cab-fd19-b96d-4bb3a2d5ec84:
[I] Jul 19 13:24:28 ndm: Opkg::Manager: init script reset to default: /opt/etc/initrc.
[I] Jul 19 13:24:28 ndm: kernel: EXT4-fs (sda1): re-mounted. Opts: (null)
[I] Jul 19 13:24:28 ndm: Opkg::Manager: /tmp/mnt/FIT8GB/ mounted to /tmp/mnt/FIT8GB/.
[I] Jul 19 13:24:28 ndm: Opkg::Manager: /tmp/mnt/FIT8GB/ mounted to /opt/.
[I] Jul 19 13:24:28 npkg: inflating "debian-keenetic.tar.gz".
[I] Jul 19 13:24:28 ndm: Core::ConfigurationSaver: saving configuration...
[I] Jul 19 13:26:10 ndm: Opkg::Manager: /tmp/mnt/FIT8GB/ initialized.
[I] Jul 19 13:26:10 ndm: Core::ConfigurationSaver: configuration saved.
[I] Jul 19 10:26:10 root: Starting Debian services...
[W] Jul 19 13:26:11 ndm: kernel: Algorithmics/MIPS FPU Emulator v1.5
[I] Jul 19 13:26:11 ndm: Opkg::Manager: /opt/etc/initrc: Starting OpenBSD Secure Shell server: sshd.
[I] Jul 19 13:26:11 ndm: Opkg::Manager: /opt/etc/initrc: Done.
[I] Jul 19 10:26:11 sshd[448]: Server listening on 0.0.0.0 port 22.
[I] Jul 19 10:26:11 sshd[448]: Server listening on :: port 22.
Using Debian
Use following SSH credentials to get access to Debian environment:
host: my.keenetic.net,
port: 222
user name: root,
password: debian.
This is it, you've got 10000+ available packages from one of greatest linux community!
For 64MB RAM devices is highly recommended to turn on Swap-file.
Details
You see chroot-environment after SSH logon, there's only minimal Busubox outside of it and /opt/etc/initrc script to fire up sandbox, this script starts necessary Debian services on boot and gracefully shut them down on reboot or unmount USB drive:
[I] Jul 19 10:28:36 root: Stopping Debian services...
[I] Jul 19 10:28:36 sshd[448]: Received signal 15; terminating.
[I] Jul 19 13:28:36 ndm: Opkg::Manager: /opt/etc/initrc: Stopping OpenBSD Secure Shell server: sshd.
[I] Jul 19 13:28:36 ndm: Opkg::Manager: /opt/etc/initrc: Done.
[I] Jul 19 13:28:36 ndm: Opkg::Manager: disk unmounted.
Also, you'll find /chroot-services.list file, where you can put new services which should be started on boot, please use one name per string, service names can be taken from /etc/init.d.
Good luck!