I'm in a similar boat. My dd is approaching ten months and a terrible sleeper (I consider a 3 hour stretch to be wonderful if I get that). I think I need to finish breastfeeding very soon (dd is waking for comfort) and instigate a bit of controlled crying with dh's help. This is going to be hard, I have done it before but in the end all our lives seemed better when we got a bit more sleep.
I made the mistake of having a late night last night and this morning I was in an awful state, added to pmt, I really felt like I wasn't coping and even wondered if pnd was setting in (I have had it twice before - I have two older children), so also have them to care for although they are school age. Back to this morning; I just felt like I was at the end of my tether and wanted to strap dd in her car seat (so that she was in a safe place) and ring someone and say come and get her. My dh works full-time and I have absolutely no extended family/other support, I try to stay strong but today for the first time, I cracked a bit and luckily dh was there to pick up the pieces.
I get how frustrating it is. I cannot do anything with dd on my lap - nothing on the computer, single handed for example, it is as if she wants my absolute undivided attention...sometimes when I have conversations with others, it is as if dd senses that attention is being diverted and she has squeals/babbles really loudly. She seems to get bored with toys (she has several different types), loves to try and eat everything around that she shouldn't (which is obviously typical).
My only saving grace is that dd has naps (although she has dropped the morning one over the past couple of days) but I can usually bank on at least one lengthy one and this literally saves my sanity.
The toddler stage is just around the corner...and it is going to mean time spent at toddler groups. My second dc was an escape artist at these places and you couldn't turn your back for one moment or they would be elsewhere like trying to pull a fire extinguisher off the wall, so no rest there and couldn't hold a proper conversation because they had to be watched.
My patience feels frayed today as you describe and I know there is more to come. I love my dd and wouldn't be without her but I get exactly where you are coming from (I should have added that I am a full-time sahm) and so it is pretty much 24/7 with a bit of light relief provided by dh. I don't want to wish time away (because it is a long way off at the moment) but I can't tell you the relief that pre-school provided when dc2 turned 3 and was able to attend...I think I am going to put dd in at the earliest opportunity even if this means paying, though I bet I will miss her terribly, she is my last dc. For me, the problem is that there is little balance in life (no-one to hand her over to during the day) and terrible sleep deprivation.
On a positive note DD has a beautiful smile and when she smiles things seem a whole lot better x Not sure that this helps, accept to say that you are not alone! If it is getting too much, it might be worth talking to someone about how you feel, I'm going to talk to the health visitor or most likely my GP if I have many more days like today.