Unless there is $60,000 of cash and a $60,000 house, then your sister will owe you money from her own funds, right?
My suggestion is that you divide the home and the liquid assets equally, so that you each own a half share of the house. You won't owe estate tax on inheriting the house or money.
Once the transfer of ownership is recorded in the county clerk's office, you can proceed to sell your share in the house to your sister for half its value. Your sister will pay you cash out of her inheritance and make up the rest of the cash from her other resources. You won't owe any income tax on the proceeds because your cost basis is equal to the value of the home on the date of your parent's death. If you sold the home for more than its inherited value, you would owe capital gains on the difference. Assuming that the value of the house on the date of death is $100,000, then your share has a cost basis of $50,000. If you sold it for $51,000, you would have a taxable gain of $1,000. But if you sell it for its inherited value, you have a zero gain and owe no tax.
It will be better to do a legal sale than a gift. Technically you can give your sister your half of the house and she can give you some money, but that will run into problems because if anyone checks, it will look like you were trying to disguise a sale and raise suspicions. It will be cleaner to simply sell your interest to her for half the value. You won't owe income tax, although you may owe a transfer and recording tax to the county (which you would also owe if it was a gift). You will probably want to have the sales paperwork processed and recorded by an attorney so you don't accidentally cause problems down the road. It should only cost a fraction of what she is going to pay you.
If your sister doesn't have the cash to pay you in full for your half of the house, you could sell her the house and take back a mortgage. She would own the house in full, and she would owe you monthly payments with interest until the balance is paid off. If you do this, the interest is taxable income but the principle is not. And if you don't charge interest because it's family, the IRS will make you pay tax as if you did charge interest (it's complicated), so get extra attorney help if you want to carry a mortgage for your sister.
