Welcome. I initially thought that a Brother 1681 was a flat bed, but upon further investigation there is also a free arm model 1681. According to the pictures I have seen the 1681 and the 1781 are very similar. There is also a Jones 1781 and a Lemair Helvetia 1781 that I believe were made by Brother.
I was able to find manuals in few places.
I am not affiliated, nor have I purchased from any of them.
http://www.devicemanuals.com/support...ual-83049.html
http://sewingwishlist.com/help/JONES-1781.html
https://www.sewingonline.co.uk/library/JONES-1781.html
and 1681 could be found at
https://www.etsy.com/listing/6972692...era-1681-jones
I'm thinking that a foot controller for a 1681 would also fit a 1781.
https://www.ebay.com/itm/Brother-Riv...8AAOSw2x1dWrVj
or
https://www.sewingpartsonline.com/fo...iant_id=281729
I also found these.
https://www.ebay.com/itm/Brother-168....c100010.m2109
As far as the extension, you may have to search a little harder, probably on ebay. There are some who have made wooden or plexiglass ones to match their machines.
I'm thinking that it may be what is called a "left-homing" machine. This means that when the machine is sewing a straight stitch the needle will come down in the left side of zigzag foot. Some machines will come with a straight stitch plate and straight stitch foot. I sometime have a hard time identifying the left homing straight stitch feet, especially without studying for a while.
I hope this helps (or maybe your aunt will find the missing items) and you can get this inherited machine back up and running smoothly.
Janey - Neat people never make the exciting discoveries I do.