One reason you were denied a home equity line of credit is because late mortgage payments put you in the category of a high-risk borrower. Your current credit score reflects that level of risk. Another reason you were denied is that you applied for a loan with the same lender using the same collateral.
Generally speaking, lenders of the same category are hesitant to lend to people who have had trouble paying an existing loan in their category. For example, you likely qualified for a high-interest auto loan because the auto lender reviewed your credit report, and although you had late mortgage payments, the lender did not see any late payments on auto loans. Had you paid any auto loans late, it is probable you might not have qualified for an auto loan at all.
Regardless, you will not qualify for competitive interest rates with any lender until your current credit score improves. How do you raise your credit score? Clean up your act! Make sure you pay all your loans on time (or early) and as agreed, every time. Don't try to get new credit unless you need it, and you can keep balances as low as possible. Better yet, don't take out another loan unless you can pay the balances in full each month.
To do this, you need to develop a workable spending plan that allows you to meet your current obligations, including your mortgage and auto loan, while saving some money each month. If you don't have substantial savings, you will fall behind every time an unexpected expense shows up. Savings is essential to financial stability, which is essential to a good credit score. You don't say why you were late with several mortgage payments, but I'll bet you had no savings to fall back on. Your goal is to never have to do so again. An emergency savings account of six to 12 months' of living expenses will allow you to dip into existing savings to pay your mortgage on time, should you not have enough funds to cover the payment in the future. So, plan to put away as much money as you can each month to fund your savings.
I know you would like to pay off your high-interest auto loan, but that may be unrealistic at this point. The incentive of getting rid of a high-interest note can motivate you to get your finances in order.