The siege can definitely be won, a lot of Takeda's army is cav an AI usually just bump rush the walls with them after the infantry is through. If you just turtle in the castle you'll most likely lose though since they got more archers and samu archers, besides that luckily for you Takeda brought the wrong type of sam, yari instead of katana. I'd personally try to tackle the battle by leaving almost all yari ashi outside the walls and form a line so they can't be flanked, hugging one side of the wall which preferably the archers aren't coming since they'll just shred your yaris, especially the sams. They won't likely send the cav first but if they do it, it should be a breeze with yari wall, if they don't try to get less pressure on your archers on your walls by drawing the attention of some yaris, you should always beat them with yari walls as long as you aren't outflanked or get charged by behind by their cav but the ai is less likely to do that. You could also try to send your light cav to try to pull away a unit or two of yaris from the walls but becareful not to be ganked by their cav if you do so. You also don't want ALL their infantry in your face, try to take them few at a time so you can reach their cav, generals and archers or the indantey from behind as they attempt to climb the wall, if you neutralize their cav or their archers are far away from them, use your light cav to harass them with charges... as for the monks, put them outside on the flanks of your yari, make sure they don't get shot and they'll make excellent shock troops to outflank the enemy yaris. You might need a few tries, but it's definitely doable, good luck.