How much does it cost to build power plant and cost of energy per kWh?

The cost of a power plant in the US is driven by regulations and indemnity (insurance) costs.

Currently, you cannot build a privately-funded power plant enywhere in the US because there is no insurance carrier that is big enough or willing to take the risk. Therefore, the cost estimates you find on the web are mostly fantasy numbers that include a "wild guess" estimation of possible insurance costs.

The construction cost alone depends on the size of a plant. The estimate of the construction cost alone is about $2B for a 1GW plant, which works out to your number of $2 per Watt or $2k/kW.

To explain the meaning of the $2k/kW number, you can assume that the plant will produce it's nominal power output, say 1GW, continuously for about 40 years. There is certainly the cost of fuel and, more importantly, the cost of processing of spent fuel, which is astronomical. But, if you ignore that, the simplified calculation is that you spend $2k and get 1kWh every hour for 40 years. If you assume the wholesale price of $0.1/kWh, the initial $2k outlay will generate $876 per year in revenue. This is not bad if you could get a free government indemnity, and had no radioactive waste disposal costs. The cost fuel fuel and the operation/maintenance of the plant would certainly take out a chunk, but these are probably relatively moderate per kW.

Keep in mind that you need to store radioactive waste for up to 12,000 years after you remove it from the reactor, so this is not a simple problem and there is no cheap solution.

The cost of a power plant in the US is driven by regulations and indemnity (insurance) costs.