Ouch!
How old is the standpipe? It doesn't look that old, but it certainly appears to be past its useful service life.
One possible short term fix:
Get some roof patching fabric, a wire wheel for your cordless drill, and roof sealant (that will hold in standing water).
After the water level in the lake drops, pump (or siphon) another foot or two of water down the standpipe. Examine the exterior for any current leaks as well as incipient leaks.
Scratch up the culvert with the wire wheel. (Judgement call on how hard to grind. You want more rust-free bare metal, but DO NOT make any large holes.)
Spread on some sealant and then press a fabric patch in the sealant to cover the hole (or holes with a single patch). Work on more sealant to the specs on the can.
Even if you don't get all of the leaks perfectly sealed, you may be able to get by with the pipe for a while longer. Especially if you pond has a lot of water inflow.
Long term fix?
Can you grout in the next smallest size of culvert and get a good seal to the lateral pipe? If so, you might be able to then just cement the smaller culvert inside the larger leaky culvert.
Bad news.
There is always a chance that the standpipe could fail catastrophically in a big rain event. If so, it could wash out the dam. Probably not dangerous on a 1-acre pond, but hopefully there is not a neighbor's house just behind the dam.
There are experts on the forum that have lots of experience replacing failed drainage pipes. Definitely listen to their advice if you can get them to comment.
Good luck on your project!
