Remote control a Roku TV from another location?

I don't know of a way to take remote screenshots of the display (maybe there's a developer mode trick?), but you can at least send remote-control keypresses and certain other commands via the ECP API, which listens for HTTP POST requests on TCP port 8060.  It must first be turned on, via

 Settings -> System -> Advanced system settings -> Control by mobile apps -> Enabled (NOT "Limited")

Also, commands normally must come from a device on the TV's local network, so you need some way to acess your MIL's LAN, such as a Raspberry Pi plugged in there and hosting a VPN endpoint, or maybe a VPN terminated directly to her router if it supports that. I wouldn't recommend port-forwarding to 8060 directly from a public IP, since the protocol isn't meant to be exposed like this, and it would open the door to others messing with her television.

Edit to add: You might want to also turn on Settings -> System -> Power -> Fast TV Start, so that the television maintains its Wifi or Ethernet connection even when "off", allowing it to also be powered on remotely.  Otherwise, your MIL would need to hit the power button herself before you could get in to do anything.  This does come at the cost of increased power consumption - about 12 watts 24/7, if I remember right, which would be 8.6 kWh/month.

Here's a simple command-line based shell script I use to control local Roku TVs from an ssh session to the router. Probably not the sort of interface you'd care for, but it might give you some ideas.

#!/bin/bash

N=`basename $0`
LASTCH=/tmp/roku-lastch-$N
SETCH=/tmp/roku-setch-$N

case "$N" in
dentv|den|dtv)
 I=10.2.1.132
 ;;
kitchentv|kitchen|ktv)
 I=10.2.1.133
 ;;
bedroomtv|bedroom|brtv)
 I=10.2.1.129
 ;;
*)
 echo "invalid basename: $N"
 exit
 ;;
esac

if [[ "$1" = "" ]]; then
 runtime $N
 exit
elif
 [[ "$1" = "$" ]]; then
  S="$2"
  c=0
  while [[ $c -lt ${#S} ]]; do
   curl -d '' http://${I}:8060/keypress/Lit_${S:$c:1}
   c=$[ $c + 1 ];
   sleep 0.1
  done
  exit
fi

FIRST=1
for k in $*; do
 if [[ $FIRST != 1 ]]; then sleep 0.2; fi
 FIRST=0
 case "$k" in
 off|on|power)
  curl -d '' http://${I}:8060/keypress/Power
  ;;
 mute)
  curl -d '' http://${I}:8060/keypress/VolumeMute
  ;;
 vol-|v-)
  curl -d '' http://${I}:8060/keypress/VolumeDown
  ;;
 vol+|v+|+)
  curl -d '' http://${I}:8060/keypress/VolumeUp
  ;;
 tuner|local|locals|tv|ota)
  curl -d '' http://${I}:8060/keypress/InputTuner
  ;;
 hdmi1|h1)
  curl -d '' http://${I}:8060/keypress/InputHDMI1
  ;;
 hdmi2|h2)
  curl -d '' http://${I}:8060/keypress/InputHDMI2
  ;;
 hdmi3|h3)
  curl -d '' http://${I}:8060/keypress/InputHDMI3
  ;;
 hdmi4|h4)
  curl -d '' http://${I}:8060/keypress/InputHDMI4
  ;;
 ch-)
  curl -d '' http://${I}:8060/keypress/ChannelDown
  ;;
 ch+)
  curl -d '' http://${I}:8060/keypress/ChannelUp
  ;;
 home)
  curl -d '' http://${I}:8060/keypress/Home
  ;;
 back|b)
  curl -d '' http://${I}:8060/keypress/Back
  ;;
 select|ok|sel|s|e)
  curl -d '' http://${I}:8060/keypress/Select
  ;;
 enter)
  curl -d '' http://${I}:8060/keypress/Enter
  ;;
 left|l)
  curl -d '' http://${I}:8060/keypress/Left
  ;;
 right|r)
  curl -d '' http://${I}:8060/keypress/Right
  ;;
 up|u)
  curl -d '' http://${I}:8060/keypress/Up
  ;;
 down|d)
  curl -d '' http://${I}:8060/keypress/Down
  ;;
 replay)
  curl -d '' http://${I}:8060/keypress/InstantReplay
  ;;
 play|pause)
  curl -d '' http://${I}:8060/keypress/Play
  ;;
 rev|rew|rewind|rr)
  curl -d '' http://${I}:8060/keypress/Rev
  ;;
 fwd|ff)
  curl -d '' http://${I}:8060/keypress/Fwd
  ;;
 info|menu|star|a|i)
  curl -d '' http://${I}:8060/keypress/Info
  ;;
 search)
  curl -d '' http://${I}:8060/keypress/Search
  ;;
 backspace|bs)
  curl -d '' http://${I}:8060/keypress/Backspace
  ;;
 findremote|find)
  curl -d '' http://${I}:8060/keypress/FindRemote
  ;;
 launch)
  curl -d '' http://${I}:8060/launch/$2
  exit
  ;;
 netflix)
  curl -d '' http://${I}:8060/launch/12
  ;;
 amazon|prime)
  curl -d '' http://${I}:8060/launch/13
  ;;
 nasa|nasatv)
  curl -d '' http://${I}:8060/launch/244655
  ;;
 media|mediaplayer|mpl)
  curl -d '' http://${I}:8060/launch/2213
  ;;
 sling|slingtv)
  if [[ $# = 2 ]]; then
   R="$2"
   curl -d '' "http://${I}:8060/launch/46041?contentId=${R}&mediaType=live"
   exit
  else
   curl -d '' http://${I}:8060/launch/46041
  fi
  ;;
 yt|youtube)
  if [[ $# = 2 ]]; then
   R="$2"
   if [[ ${R:0:4} == "http" ]]; then
    V="`echo $R | sed 's/.*v=//; s/&.*//'`"
   else
    V="$R"
   fi
   curl -d '' "http://${I}:8060/launch/837?contentId=${V}&mediaType=live"
   exit
  else
   curl -d '' http://${I}:8060/launch/837
  fi
  ;;
 apps)
  lynx -dump http://${I}:8060/query/apps
  ;;
 active-app)
  lynx -dump http://${I}:8060/query/active-app
  ;;
 tv-channels)
  lynx -dump http://${I}:8060/query/tv-channels
  ;;
 tv-active-channel)
  lynx -dump http://${I}:8060/query/tv-active-channel
  ;;
 *)
  if [[ $k =~ ^[0-9.-]+$ && $# = 1 ]]; then
   if [[ "$k" = "-" ]]; then
    if [[ -f $LASTCH && -f $SETCH ]]; then
     C=`cat $LASTCH`
     mv -f $SETCH $LASTCH 2>/dev/null
    else
     echo "No previous channel set."
     exit
    fi
   else
    mv -f $SETCH $LASTCH 2>/dev/null
    C="$k"
   fi
   echo $C >$SETCH
   if [[ "${C:1:1}" != "." && "${C:2:1}" != "." ]]; then C=${C}.1; fi
   echo "channel: $C"
   if [[ ${#C} -lt 4 ]]; then C=0$C; fi
   L=$[ ${#C} - 1 ]
   for d in `seq 0 $L`; do
    curl -d '' http://${I}:8060/keypress/Lit_${C:$d:1}
    sleep 0.1
   done
  else
   echo "literal $k"
   curl -d '' http://${I}:8060/keypress/Lit_$k
  fi
  ;;
 esac
done

Hmm, a few minutes ago I made a longish post here explaining how to do this using ECP protocol, including a code sample (just a bash shell script) and discussion of security considerations, but the reply seemingly vanished!

Did a moderator delete it?  If so, why?

I did label the script as Python code because Unix shells (Bourne/bash etc.) were not amongst the options, and that seemed closest. Figured that question was purely for syntax highlighting & such.