This is a looooong reply, but I hope something in it helps.
Sally W's response was great! There is nothing wrong or inappropriate about children nursing for comfort. That is part of what nursing is all about! Your daughter's need for you is valid, and nursing is a valid way to meet that need. You mentioned her nursing increasing. Is she going through any changes in her life right now that might be making her feel insecure? Often children need extra comfort when they are learning a new skill, starting daycare, moving, are sick, are teething, when Daddy is working late hours, or any other times when things are changing in their lives. Also, she may be going through a growth spurt, and honestly needs more nutrition. Despite urban legends to the contrary, your milk stays rich in nutrients and immune factors well past baby's first year. In fact, these components are actually more concentrated in your milk in the second year of baby's life! Kellymom.com has many articles about weaning issues at http://www.kellymom.com/bf/weaning/index.html
One of my favorite things about nursing a toddler is that is is a much more 2-sided relationship than nursing a small baby. You have the freedom to consider your own needs, to delay or refuse nursing if you need to. You can start using words with your daughter, naming her feelings and explaining your own. There is a lot more freedom in nursing a toddler than nursing a newborn. You may find that by making some changes, you are ready to continue nursing until your daughter weans voluntarily. You may also find that you are ready to wean now. The best thing you can do is to be as responsive to your daughter's needs as you can, and to also respect your own needs.
I do want to say that you are NOT the only mom nursing a toddler. I nursed my first until she was 21 months; even then we only stopped because I was pregnant and it started to drive me nuts. Without the pregnancy, I would likely have continued. My second is happily nursing still at 15 months. So, if you want to stop because you think it's weird or that you are the only freaky person nursing a walking person, that is not true. One thing that really gave me stamina to continue nursing was to night wean. We simply told our daughters that “Mommy goes night-night, Daddy goes night-night, baby goes night-night, and milky goes night-night.” If she asked to nurse or fussed, I reminded her that milkies went night-night, and she could nurse in the morning.
You mentioned having problems with your daughter demanding to nurse, especially her throwing a fit and pulling at your shirt. This is a issue of manners, and now is the perfect time to start teaching them. Like any other manners, nursing manners must be taught. Both my children were shirt grabbers, and my second is really bad at it. What we did/do is to gently but firmly take hold of her wrist when she pulls at my shirt and say "No pulling Mommy's shirt. Say 'milk please.'" Then we demonstrate the signs for milk and please. I do not allow her to nurse until she asks politely. The crying isn't about being demanding, it's about being disappointed. Little ones don't have an appropriate mechanism for expressing disappointment, especially at 16 months old. Now is a great time for starting to name your daughter's feelings for her. We say something like "I see you crying. You are disappointed because you are not getting milk. You are frustrated because you want milk right now. You need to ask politely for milk. We say 'milk please' [demonstrating the signs to her] when we want milk."
If you are concerned about nursing in public, I completely understand. I, too try to avoid nursing my 15 month old in public, because there are just too many rude people out there. Some tips for avoiding it... nurse before you leave, and bring a snack and sippy cup with you. If she asks to nurse while you are out, just offer the snack/sippy and then tell her "We have milk at home." If you do this, you HAVE to stick to it and only have milk at home. Some people also nurse only in a certain chair, or only lying down on the bed, so that your toddler knows she cannot nurse anywhere else. You could also have a special toy that she only gets when you guys are out of the house.
Of the two books recommended, I personally preferred "How Weaning Happens." I felt like much less pressure from it to continue nursing beyond the point that I was ready to stop.
I know this can be a tough time. You have done wonderfully for your daughter by nursing her this long. Whether you decide to continue nursing or to wean, the best thing you can do is to keep being responsive to your daughter's needs.
Best of luck,