Hi A.,
My younger son also refused a bottle or sippy cup for breastmilk (and I was working outside the home). You don't mention what sort of health issue your older child has, but have you considered that your younger child may have a similar health issue and that the breastfeeding provides comfort. My younger son nursed for a very long time and we only later learned (when he was 8) that he (and older brother and myself) have celiac disese (an intolerance to wheat gluten that can cause a myriad of health issues) and a dairy allergy. You don't mention how frequently you nurse during the day or how busy your day is, but it could be that he's not getting the time he needs during the day, so he's wanting to nurse more at nite. I know it's difficult when you have two, but I found that nursing during the day while reading books with the older child worked well to ensure that both their needs were met. I know it is exhausting being up at nite and what we found worked best for all of us was to bring the nursing baby's crib into our room and put it right beside our bed or just bring him to bed with us. You might also want to consider whether something your eating (e.g., dairy) or drinking (e.g., caffeine) might be resulting in the nighttime wakefulness or causing him discomfort that he is awakening so frequently at nite. For my oldest son, in hindsight it was likely the dairy in my diet that caused him to be a poor nite time sleeper. My second son slept quite well at nite at that age (he would only awaken once for nursing), but I had removed dairy from my diet by that time.
I wish you well - I know it is frustrating, but these sort of demands from your little ones only last such a short time before they're replaced by a whole new set and type of demands. Given that you are feeling so exhausted, you might want to have you doctor check out your thyroid. If you have low thyroid function, that can impact your milk supply and might be why your son wants to nurse so frequently because he's not getting as much in each feeding and low thyroid function obviously increases your fatigue. Also, you might consider getting a mother's helper in (maybe a teen in the neighborhood for an hour or so each day) to give you some time away from the kids even if you stay home and just take a relaxing bath - without your husband there 5 days a week, you are carrying a huge burden. Also, you might dedicate some daddy time on the weekend to give yourself some time alone. It sounds like you also have a daytime separation issue. Your 13 month old should be able to go without you and be given some solid foods/snacks etc. With my younger son, if I was there, he certainly wanted to breastfeed, but when I wasn't around, he would take water or juice from a sippy cup (just not breastmilk). You really need to enlist dad's help on the weekends to resolve these issues. What we did with our oldest son when he was 15 mos old and waking to nurse many times at nite was we had Dad give him water in a sippy cup - my son quickly learned that it wasn't worth waking up for just water. But, I made sure I was available to nurse when he needed to during the day, as well.
Good luck and just feel assured that you really are doing the best you can and that by breastfeeding you are really providing for you son both nutritionally and emotionally during these very important formative years.