Step #1: Talk to your Landlord
The lease you signed was most likely provided by your landlord. If there is a provision regarding what happens if you break the lease early, it's going to be written in your landlord's favor. In court, you aren't going to be given much leeway when it comes to a contract that you have signed, so its always a good idea to see if you can work something out with your landlord first. If they'll agree to waive the lease-break fee or to another deal that you can live with, then you will save a lot of time and money. Make sure to get it in writing if they agree to some sort of deal.
Step #2: Give Notice
Even when you break a lease, your landlord still has an obligation to find a new renter. However, your landlord can't start looking for new tenants until they know you are leaving. Give notice in writing and the sooner you do it, the better.
Step #3: Get Someone to Take Over your Lease
Try to find someone who wants to move into your place for the remainder of your lease. Introduce potential tenants to your landlord. In the best case scenario, your landlord will accept the new tenant and they will then take over your lease. At worst, you'll at least have evidence that there were other interested tenants if your landlord later tries to sue you for a large lease-break fee or months of rent that could have been paid by another tenant.
Step #4: Potential Legal Arguments
If things go poorly and your landlord ends up suing you for a lease-break fee or lost rent, you have several potential legal arguments. First, if the lease said nothing about a lease-break fee, then your landlord cannot make up a fee on their own. Instead, you are only liable for money lost as a result of you breaking the lease. Second, your landlord has a duty to try to find a new tenant for the remaining lease period. If your landlord refuses to even try to find a new tenant, it is not fair for you to pay for months of missed rent or a large lease-break fee. Third, your landlord can only recover their actual losses. If rent is $1,000/month, and the lease-break fee is $2,000, and the landlord re-rents the place within 15 days, then it would be unfair to recover the entire $2,000 lease-break fee. To allow this would be to allow your landlord to recover their money twice (once from your lease-break fee, and then again from the new tenant).