Old thread but, in case, the OP still wants a shortcut or someone else has this question in the future, you can record a macro in Word that deletes the line. (Tools — Macros — Record Macro) Name it something like delete_line. (No spaces allowed in the name of macros.)
After recording, go back to Tools, Macros, and click Stop Recording.
Then, you can assign the macro to a keystroke of your choice. (Tools — Customize Keyboard and scroll down the left menu to Macros)
Command-delete will then delete from the cursor point to the end of the line.
If you want to delete the entire line, you'd want to start the macro with option-left arrow — that’s the keystroke that moves the cursor to the start of the line. Then, do the command-shift-right arrow biz to highlight the line.
Many of us, though, rarely delete a line at a time and find it just as convenient to move the mouse pointer to the left margin next to the line, click there, thus highlighting the line, and press delete. I use Customize Keyboard to assign Command-Delete to DeleteWord, which deletes the word ahead of the cursor, a carry over from PC Write or Apple Writer days.