In addition to the information given by Andrzej A. Flip (from https://superuser.com/questions/1254358/replace-single-line-with-multline-content-within-the-file-nginx-conf/1254557#1254557 ), you need to escape the semicolon (;), asterisk (*), and forward slash (/) characters. The \n makes it a new line, and use -i.bak to make a backup of the file before making the change with an .bak extension.
sudo sed -i.bak 's/^ include \/etc\/nginx\/sites-enabled\/\*\;$/&\n \n server {\n listen 8081\;\n \n location \/ {\n proxy_pass http:\/\/127.0.0.1:8080\/\;\n proxy_redirect default\;\n auth_basic "Server Administration"\;\n auth_basic_user_file \/etc\/nginx\/passwords\;\n }\n}\n/' /home/pi/Public/NGINX-test/nginx.conf
sudo sed -i.bak 's/^ include \/etc\/nginx\/sites-enabled\/\*\;$/&\n \n server {\n listen 8081\;\n \n location \/ {\n proxy_pass http:\/\/127.0.0.1:8080\/\;\n proxy_redirect default\;\n auth_basic "Server Administration"\;\n auth_basic_user_file \/etc\/nginx\/passwords\;\n }\n}\n/' /home/pi/Public/NGINX-test/nginx.conf