10K is 10k no matter what the wattage.
When in doubt, go bigger.
The wattage rating is just how much power it can handle, the circuit does not care.
As long as it is a high enough wattage to handle the load, it does not matter how much bigger it is.
When dealing with resistors, there are a couple of other things that will sneak up and bite you.
In this case, they don't matter, this is just for future reference.
One is precision. Many cheap carbon resistors are +/- 20%.
But nowadays, most of them are like yours +/- 5%. Which means the resistor can be anywhere from 10,500 ohms to 9,500 and still be correct.
If the circuit is critical, you can buy 1% ones.
The other problem you can run into concerns wattage again.
1 Watt is as big as you can normally find in carbon resistors. There are some 5 watters out there, but they are rare and usually special order.
In general a resistor above 1 watt is wirewound.
A carbon resistor is made out of carbon. Its thickness determines the ohms.
When alternating current flows through a piece of carbon, in general it just sees a resistance.
But, a wirewound resistor uses resistance wire wound around a form.
DC does not notice, but when higher frequency current tries to go through a coil like that, you get into inductance. Which can cause all sorts of problems.
So, as long as you are working with DC, don't worry. But if you ever want to replace a resistor in an AC circuit, make sure to replace it with the same type resistor.