CORPS (available only in PDF) - tactical combat is hex based, but can be done more narratively fairly easily. Uses only 1d10 per player, and that seldom; autosuccess is used a lot. Works really well, but a bit mathy for some.
Castle Falkenstein is a card based victorian steampunk fantasy, It uses standard bridge or poker decks, and works just fine with separate decks per player; magic users need to share a single magic deck (which is still just a poker or bridge deck), but that is not used all that frequently. Further, the character sheet is a journal; inexpensive journals can be used (and reused), and preclude needing a clipboard. Pencil is adequate, so writing upwards isn't an issue either.
Active Exploits is a passable diceless system; I'm not fond of it, but it has plenty of fans.
Theatrix is a diceless system with optional d100 mechanics. I like it fairly well, but it's sadly out of print. Caveat: I've only run it using the dice based mechanics.
TSR released two card based games (Dragonlance 5th Age, and Marvel Super Heroes Adventure Game), each using custom decks. Both are out of print. Both are labeled with the "Saga System" logo (which shouldn't be confused with Star Wars Saga Edition, which is derived from the d20 system).
Coping with Prone
Note that for die-rolling, a cheap, lightweight mirror and a shallow box can be used to make the rolls readable prone. I've run games while prone (recovering from knee surgery), and I used a laptop and a die-rolling program, but I've had friends who rolled into a box and the read the dice using a mirror.
Any game using d10's only can be played with a poker deck, with all the faces removed. d12's only can be done as well, by removing one face (I'd pull the kings, and read J as 11 and Q as 12). As mentioned for Castle Falkenstein, cards are easily handled prone.
If one's willing to make custom decks, a deck marked 1-12, with a smaller 1-6 can be used for replacing both d12 and d6 (and by extension, d3). At least 4 sets (48 cards) would be good; 5 sets allows adding d10 markings, but also means every card is unique
Maps and minis can be easily worked around. There's this neat product called Blu-Tac; generic versions are called poster putty or poster tac. Use it to put a laminated map on the TV screen. Use it to put flat counters on same said map. Or, use a map on poster board (laminate it) and hang it in her eyeline at a 45° angle, again using blu-tac to hold counters on the map.
The Fiery Dragon token sets work GREAT for this - they're cardboard, about 1mm thick, and pretty clear, and consistently 1" diameter.
If you have the money, you can buy sheet magnets or roll magnet, often peel-n-stick. Some whiteboards are ferrous (will hold a magnet); sheet steel can also be purchased, then gridded, the edge rolled or sanded and taped. Counters can be mounted to sheet or roll magnet, and stuck to the board.