1) You create directory .ssh ( but why -p?) only what permissions has it ? shoudl be read only for the user...
2) Your cat command will also ask you tha password since the key has not been transfered yet.. and that method may not be all that safe of corruption.. the safest would have been to copy the file to add .A extension, then scp that file to server B in her .ssh directory, then update authorized_keys file