The datasheet is here: http://pdf.datasheetcatalog.com/datasheet/toshiba/3835.pdf
You can see the shunt resistor (which you will be replacing) as Rs on p.4.
In the table on p.3, you can see that the Detection Voltage depends on the overall voltage, but the way it works is that when the voltage is below that level, the flasher goes to hyperflash mode. As you can see in the table, the detection voltage is higher for higher operating voltages. If the car is running, Vcc = 14.4 ~= 15V. From the table, the minimum detection voltage is 87 mV. At 12V operating voltage, the minimum detection voltage is 75 mV.
So you want to pick a resistor that will give no less than ~75 mV drop across the resistor when both bulbs on a side are good, and less than that when only one bulb is good.
Now we need to know info about your LED. I believe mine used 350 mA each. When you turn, you will be using one in front and one in back. So that's 700 mA for my LEDs. A 0.15 Ohm resistor creates a voltage drop of V = I*R = 0.700 * 0.15 = 0.105 V = 105 mV. That means the module will think that the lights are okay.
If one bulb fails, the current drops to 350 mA, giving a voltage drop of 0.350 * 0.15 = 52.5 mV, which is less than the detection voltage and hyperflashing would be triggered.
One thing to be careful about is the power dissipation. P = I^2 * R. So here, using the LEDs, P = 0.700 * 0.700 * 0.15 = 0.07 Watts. With Hazard lights on, you use all four bulbs, so the power goes up to 1.4^2 * .15 ~= 0.3 W. That means you need a 1/2 Watt resistor. Although, as mcrider pointed out in his very helpful writeup, you should have more of a margin for safety. He chose a 1W resistor. I actually used a 3.75W resistor, because it wasn't much more expensive and it seemed good. I used this one.
If you use incandescent bulbs with this modified flasher, you will have a current of about 9 Amps, so power dissipation of about 12 Watts. Poof goes the resistor and maybe starts a fire and the car burns to the ground. So please be sure to only use LEDs with your modified flasher module, and only the LEDs you used to calculate the appropriate resistor. (At least check the calculation if you change to different bulbs.)
Let me know if something didn't make sense. I chose the resistor that I went with, which was pretty easy to determine with a spreadsheet so I could try different values.
