I am not sure what you consider "enduro" to be. To me, an enduro is a type of race. In fact, it is a type of race that puts the most demands on a bike. For now I am going to assume that what you need is a trail bike.
You certainly need to get off that TT-r125, you are way too tall for it. Having your center of gravity that high on small wheels and short wheel base is just asking for a trip over the handlebars.
You said you didn't want to go over a 250: I am guessing it was actually a 230.
Dirt bikes come in several different flavors, but the two major classifications are what I call "MX" and "trail" (trail is sometimes called "off road")
The MX bikes are built for racing. They have high performance engines which often require high octane (premimum) gas, they have longer travel suspension, and they DON"T have any frills such as kickstands or headlights. If it doesn't help the bike get around the motocross track, it isn't on the bike. These bikes are much lighter and faster than their trail bike cousins. The high performance engines are also higher maintenance, often requiring top end rebuilds after as little as 20 hours of use.
The trail bikes are built for non-competitive pleasure riding. They have lower compression engines that run just fine on the cheap gas. They don't use exotic metals to reduce weight so they are heavier but also much less expensive. They usually include a kickstand and frequently include electric start (which adds a lot more weight). Trail bikes tend to max out at 230cc.
A full size MX bike will have a seat height of around 38 inches. It doesn't matter if it is a 125 two stroke or a 450 four stroke, the frame size is essentially the same. The "mini" MX bikes vary in size and different manufactures create a niche by having a bike that is slightly different size than the other manufacturers. Someone 5'0" tall might fit a Kawasaki best while someone 5'2" tall might fit a Honda best.
The trail bikes are generally a little shorter, with a maximum seat height of around 34 inches. For reference, that TT-R125 would have had a seat height of either 30.5" or 31.7" depending on if it was the "L" model (19" front wheel, 16" rear vs 17"/14").
The biggest problem with selecting a bike for you is that you are "in between". A bike that is best for you right now won't be best for you in a year or two. I recommend getting a bike that suits you best now and plan on upgrading again in a year or two.
If you had the experience I would suspect that you would be able to handle a full size MX bike as I know a number of short adults who ride them. My daughter is only 5' 2' and she is comfortable on a 34" seat height so you should be able to stretch another 4 inches. Until you get the experience, however, having a bike that allows you to put both feet solidly on the ground is going to help you out a lot.
If you can find one the bike that I would consider to be the best is a Yamaha TT-R250. NOT a 230, a 250. Be careful, some people refer to their 230 as a 250 when it isn't. The TT-R250 is more of a MX bike than a trail bike: it has the high compression engine and long travel suspension but has a 35" seat height. I haven't ridden one, but based on the specifications I would think this bike would be perfect.
The TT-R250 is hard to come by, so.....
Considering your budget, skill level, and mechanical skills I would recommend a Yamaha TT-R230 or a Honda CRF230F (pay close attention to the suffix letters on Hondas, there is a big difference bewtween and "R" and "F".) These bikes sold for about $4K new so used ones that are only a few years old should be in your price range. Being only a few years old they should be in pretty good shape and not require much work. The engines are very reliable and can run a long time with minimal maintenance (Change the oil often!)
Rod