Do you have a parenting plan? You and the mother should work out a parenting plan, in writing, that establishes a pattern for you to spend time with your daughter. A parenting plan does a lot to reduce the stress between you two. Children of that age respond well to routine (it is easier for your daughter if she goes with you at a regular set time). And it will take away the dynamic where you have to plead to spend time with your child.
Usually fathers get to have their kids every alternate weekend at least. This seems to be the norm these days. And, you probably have a legal right to insist on this.
Your exact legal rights depend on the state you live in and circumstances of the relationship (i.e. whether you were married etc.) Of course, it is much better to work things out amicably with the mother... but you can insist on having a reasonable amount of time with your child.
If you feel it would help, you can see a lawyer (it wouldn't hurt to talk to one in any account and the mother doesn't even have to know if you decide not to go forward with this).
Of course, as I keep saying, you can also get a counselor that specializes in children of divorce. This might be a much less contentious way to work it out with the mother than going straight to the legal system.
I think you should insist on having a reasonable amount of time to build and maintain a relationship with your child. Your daughter deserves to grow up having a healthy relationship with her father.