Firm's have more than one time of debt, in the form of short-term and long-term. Both can accrue interest but not all firms present individual interest rates for ALL liabilities.
If I took 'interest expense' and used along with effective tax rate, to find 'after tax interest expense' then divided this by total liabilities and multiplies by 100. This would give me interest rate...

However, this would be assuming ALL liabilities accrue interest and it would be an average of rate at which each accrues interest.
When using this as part if Discount Rate formula, would it be representative enough??

Any better ways, if not?

using such an approach can only give you an average and I don't understand why tax is in the equation. To get the average interest rate you need to know the total debt liability for both the opening balances and the closing balances which as you say will comprise both short term and long term debt you also really need to know if debt was retired during the year and added during the year as you want to be able to calculate the current rate. Most corporations as part of their annual report provide schedules of interest bearing debt. You have to be aware that some liabilities may not be interest bearing as you have said

The question is what are you using this information for?