It is likely the other disk has issues as well which is causing the rebuild to fail. I would suggest the first thing to do is to try to create a disk image from the existing disk. It's best to use a cdrom or usb stick to boot knoppix or something.
Then assuming the drive is located at /dev/sda and you have space available on a different (USB mounted) drive mounted at /mnt/data run something like:
dd bs=200M conv=noerror if=/dev/sda of=/mnt/data/sda.img
I use bs=200M to speed up the copy, it will copy 200MB chunks at once, I use conv=noerror to allow dd to continue if it finds a disk error. See the dd man page for more information.
If you want to use ssh to copy the image over while it is created see: https://unix.stackexchange.com/questions/132797/how-to-dd-a-remote-disk-using-ssh-on-local-machine-and-save-to-a-local-disk
If dd is not working and throws too many errors the disk is close to being unsalvagable and you want to try ddrescue or something: https://help.ubuntu.com/community/DataRecovery#Imaging_a_damaged_device.2C_filesystem_or_drive
To create a bootable disk out of that image find a disk of equal or bigger size and connect it through USB for example. Let's say the new disk is at /dev/sdd:
dd bs=200M conv=noerror if=/mnt/data/sda.img of=/dev/sdd
Once done, shut down the system and replace the existing raid disk with the new one and boot the system to see whether it boots. You may have to (re)-configure the raid controller, not sure. It may boot into read-only mode again, since the raid is still degraded.
Now insert another disk and allow the raid to rebuild. It may now finish successfully. It is possible the raid controller will see the disk as a JBOD and you will need to add it again into a raid before you can have a fully working raid1.