Nichrome melts at around 1400°C according to Wikipedia, so you'll need careful control to get the kiln itself to 1300°C, including (I suspect) ramping down the power as the kiln gets close to temperature. Kanthal isn't much better (1425°C, going to 1500°C for some grades in some applications).
You could use 100% duty cycle at first, then reduce it steadily as the kiln warms up.
Don't forget that when the air is only a little cooler than the element, heat transfer will be reduced, leading to a hotter element for the same input power. The same is true for heat transfer into solid parts of the kiln. A hotter element will have a higher resistance, reducing the power at constant voltage, but this is a fairly weak effect in nichrome. If you have a suitable ammeter then (assuming the input voltage is constant), you should be able to estimate the element temperature from the resistance, i.e. from the current. There are simpler, more accurate, and more expensive ways to measure the element temperature of course.
But switching over a timescale of seconds is likely to be too long. Think how fast a light bulb comes up to full brightness as an indicative timescale. In 4 seconds it's probably starting to melt already, and it probably only survives a few such cycles. But switching with a relay much faster than a second will soon wear out the relay. Naively I'd be looking at either:
- stage lighting dimmer hardware - the same as household dimmer switches but rated to several kW.
- solid state relays, but I've only used little ones. They'll switch faster than mechanical relays, and don't wear out when used within spec - but they do dissipate some heat of their own as they toggle so you'll need to take care they don't burn out.
Otherwise I'd look at ramping down the duty cycle as things start to heat up, from 100% below about 1000°C to probably only about 10% with a period of 1s at 1300°C. This slower heating will also deal with another issue - your thermometer reading probably lags behind the air temperature nearest the element as heat doesn't distribute itself instantly. Note that your ramp rate may not suffer too much, as it's probably already off for long periods - it's just that the periods it's on for are too long.
If it came with a controller, all this should be built in, for some value of "should".