When do you use cost of equity (KE) and weighted average cost of capital (WACC) in valuation of a company?

It depends on the capital structure of the company.

Cost of equity (KE) is used when an enterprise is funded by equity capital only whereas WACC method is used where a company has been substantially funded  through debt in its caopital structure.

When capital structure consists of both equity and debt we use WACC, and if project or company is wholly financed with equity we use the cost of equity.